Delay-Aware Multi-Path Routing in a Multi-Hop …...Delay-Aware Multi-Path Routing in a Multi-Hop...

Delay-Aware Multi-Path Routing in a Multi-Hop Network:Algorithms and Applications

Qingyu Liu

Dissertation submitted to the Faculty of theVirginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophyin

Computer Engineering

Haibo Zeng, ChairMichael S. HsiaoJung-Min Park

Hesham A. RakhaYaling Yang

May 9, 2019Blacksburg, Virginia

Keywords: Delay-Aware Network Flow, Delay-Aware Multi-Path Routing, Multi-HopNetwork, Network Resource Allocation, Polynomial-Time Approximation Algorithm

c© Copyright 2019, Qingyu Liu


Qingyu Liu

(ABSTRACT)

Delay is known to be a critical performance metric for various real-world routing applicationsincluding multimedia communication and freight delivery. Provisioning delay-minimal (or atleast delay-bounded) routing services for all traffic of an application is highly important. As abasic paradigm of networking, multi-path routing has been proven to be able to obtain lowerdelay performance than the single-path routing, since traffic congestions can be avoided.However, to our best knowledge, (i) many of existing delay-aware multi-path routing studiesonly consider the aggregate traffic delay. Considering that even the solution achieving theoptimal aggregate traffic delay has a possibly unbounded delay performance for certainindividual traffic unit, those studies may be insufficient in practice; besides, (ii) most existingstudies which optimize or bound delays of all traffic are best-effort, where the achievedsolutions have no theoretical performance guarantee.

In this dissertation, we study four delay-aware multi-path routing problems, with the delayperformances of all traffic taken into account. Three of them are in communication andone of them is in transportation. Note that our study differ from all related ones as weare the first to study the four fundamental problems to our best knowledge. Although weprove that our studied problems are all NP-hard, we design approximation algorithms withtheoretical performance guarantee for solving each of them. To be specific, we claim thefollowing contributions.

Minimize maximum delay and average delay. First, we consider a single-unicast settingwhere in a multi-hop network a sender requires to use multiple paths to stream a flow ata fixed rate to a receiver. Two important delay metrics are the average sender-to-receiverdelay and the maximum sender-to-receiver delay. Existing results say that the two delaymetrics of a flow cannot be both within bounded-ratio gaps to the optimal. In comparison,we design three different flow solutions, each of which can minimize the two delay metricssimultaneously within a (1/ε)-ratio gap to the optimal, at a cost of only delivering (1 − ε)-fraction of the flow, for any user-defined ε ∈ (0, 1). The gap (1/ε) is proven to be at leastnear-tight, and we further show that our solutions can be extended to the multiple-unicastsetting.

Minimize Age-of-Information (AoI). Second, we consider a single-unicast setting where in amulti-hop network a sender requires to use multiple paths to periodically send a batch of datato a receiver. We study a newly proposed delay-sensitive networking performance metric, AoI,defined as the elapsed time since the generation of the last received data. We consider theproblem of minimizing AoI subject to throughput requirements, which we prove is NP-hard.We note that our AoI problem differs from existing ones in that we are the first to considerthe batch generation of data and multi-path communication. We develop both an optimal

algorithm with a pseudo-polynomial time complexity and an approximation framework witha polynomial time complexity. Our framework can build upon any polynomial-time α-approximation algorithm of the maximum delay minimization problem, e.g., the one from [64]with α = 1 + ε given any user-defined ε > 0, to construct an (α + c)-approximate solutionfor minimizing AoI. Here c is a constant dependent on throughput requirements.

Maximize network utility. Third, we consider a multiple-unicast setting where in a multi-hopnetwork there exist many network users. Each user requires a sender to use multiple pathsto stream a flow to a receiver, incurring an utility that is a function of the experienced max-imum delay or the achieved throughput. Our objective is to maximize the aggregate utilityof all users under throughput requirements and maximum delay constraints. We observethat it is NP-complete either to construct an optimal solution under relaxed maximum delayconstraints or relaxed throughput requirements, or to figure out a feasible solution with allconstraints satisfied. Hence it is non-trivial even to obtain approximate solutions satisfyingrelaxed constraints in a polynomial time. We develop a polynomial-time approximation al-gorithm. Our algorithm obtains solutions with constant approximation ratios under realisticconditions, at the cost of violating constraints by up to constant-ratios.

Minimize fuel consumption for a heavy truck to timely fulfill multiple transportation tasks.Finally, we consider a common truck operation scenario where a truck is driving in a nationalhighway network to fulfill multiple transportation tasks in order. We study an NP-hardtimely eco-routing problem of minimizing total fuel consumption under task pickup anddelivery time window constraints. We note that optimizing task execution times is a newchallenging design space for saving fuel in our multi-task setting, and it differentiates ourstudy from existing ones under the single-task setting. We design a fast and efficient heuristic.We characterize conditions under which the solution of our heuristic must be optimal, andfurther prove its optimality gap in case the conditions are not met. We simulate a heavy-duty truck driving across the US national highway system, and empirically observe thatthe fuel consumption achieved by our heuristic can be 22% less than that achieved by thefastest-/shortest- path baselines. Furthermore, the fuel saving of our heuristic as comparedto the baselines is robust to the number of tasks.


Qingyu Liu

(GENERAL AUDIENCE ABSTRACT)

We consider a network modeled as a directed graph, where it takes time for data to traverseeach link in the network. It models many critical applications both in the communicationarea and in the transportation field. For example, both the European education network andthe US national highway network can be modeled as directed graphs. We consider a scenariowhere a source node is required to send multiple (a set of) data packets to a destination nodethrough the network as fast as possible, possibly using multiple source-to-destination paths.In this dissertation we study four problems all of which try to figure out routing solutionsto send the set of data packets, with an objective of minimizing experienced travel time orsubject to travel time constraints. Although all of our four problems are NP-hard, we designapproximation algorithms to solve them and obtain solutions with theoretically bounded gapsas compared to the optimal. The first three problems are in the communication area, andthe last problem is in the transportation field. We claim the following specific contributions.

Minimize maximum delay and average delay. First, we consider the setting of simultaneouslyminimizing the average travel time and the worst (largest) travel time of sending the set ofdata packets from source to destination. Existing results say that the two metrics of traveltime cannot be minimized to be both within bounded-ratio gaps to the optimal. As acomparison, we design three different routing solutions, each of which can minimize the twometrics of travel time simultaneously within a constant bounded ratio-gap to the optimal,but at a cost of only delivering a portion of the data.

Minimize Age-of-Information (AoI). Second, we consider the problem of minimizing a newlyproposed travel-time-sensitive performance metric, i.e., AoI, which is the elapsed time sincethe generation of the last received data. Our AoI study differs from existing ones in that weare the first to consider a set of data and multi-path routing. We develop both an optimalalgorithm with a pseudo-polynomial time complexity and an approximation framework witha polynomial time complexity.

Maximize network utility. Third, we consider a more general setting with multiple source-destination pairs. Each source incurs an utility that is a function of the experienced traveltime or the achieved throughput to send data to its destination. Our objective is to maximizethe aggregate utility under throughput requirements and travel time constraints. We developa polynomial-time approximation algorithm, at the cost of violating constraints by up toconstant-ratios. It is non-trivial to design such algorithms, as we prove that it is NP-complete either to construct an optimal solution under relaxed delay constraints or relaxedthroughput requirements, or to figure out a feasible solution with all constraints satisfied.

Minimize fuel consumption for a heavy truck to timely fulfill multiple transportation tasks.

Finally, we consider a truck and multiple transportation tasks in order, where each taskrequires the truck to pick up cargoes at a source timely, and deliver them to a destinationtimely. The need of coordinating task execution times is a new challenging design space forsaving fuel in our multi-task setting, and it differentiates our study from existing ones underthe single-task setting. We design an efficient heuristic. We characterize conditions underwhich the solution of our heuristic must be optimal, and further prove its performance gapas compared to the optimal in case the conditions are not met.

v

Dedication

To my parents and my wife.

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Acknowledgments

I would like to acknowledge my advisor, my collaborators, members of my PhD committees,and my family for their continuous support during my five-year PhD journey. Without them,I cannot defend my dissertation.

First and foremost, I feel greatly lucky to be advised by Prof. Haibo Zeng. I almost knownothing about how to do research and become a good PhD before meeting him. He spendscountless hours with patience on helping me find the important and right research directions,work out technical details, and present papers. In addition to doing research, his help alsogreatly influences many other aspects of my life, e.g., he helps me improve my communicationskills and teaches me how to prepare professionally for interviews. I cannot thank himenough. He is a lifelong mentor for me, and is the researcher that I strive to become.

I want to sincerely thank Prof. Minghua Chen for advising me throughout all my PhDyears, and for hosting me when I visited The Chinese University of Hong Kong in fall2017. Although he works at Hong Kong which is far away from Virginia Tech, he helpsme frequently through online discussions. His suggestions are invaluable for all of my PhDresearch projects. I really appreciate his guidance, support, and encouragement. I feel veryfortunate to work closely with him.

I would also like to acknowledge the helpful feedback from the members of my PhD com-mittee: Prof. Jerry Park, Prof. Hesham A. Rakha, Prof. Michael Hsiao, and Prof. YalingYang. I benefit a lot during many insightful discussions with them, and their comments onmy work have helped me to improve the quality of my dissertation.

Finally, no words can express my gratitude to my wife Xiaoxin An, my mother ChanghuaShi, and my father Shanting Liu. Without their encouragements, I will not even considerpursuing a PhD degree. I love you forever, my dear wife and dear parents.

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Contents

1 Introduction 1

1.1 Motivation and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Contributions and Dissertation Structure . . . . . . . . . . . . . . . . . . . . 4

2 Simultaneously Minimize Maximum Delay and Average Delay with Through-put Requirements 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7


2.3 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Two Delay Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Three Delay-Aware Network Flow Problems . . . . . . . . . . . . . . 12

2.3.3 Minimizing the Two Delay Metrics by Sacrificing Flow Rate . . . . . 13

2.4 A Flow-Sacrificing Algorithm to SO . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Obtaining fSF[(1− ε)R] in a Polynomial Time . . . . . . . . . . . . . 14

2.4.2 Critical Delay Properties of fSF[(1− ε)R] . . . . . . . . . . . . . . . . 15

2.5 Constant Bi-Criteria Delay Gaps . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.1 Comparing fSO[(1− ε)R] with the Optimal . . . . . . . . . . . . . . . 16

2.5.2 Comparing fSF[(1− ε)R] with the Optimal . . . . . . . . . . . . . . . 17

2.5.3 Comparing fNE[(1− ε)R] with the Optimal . . . . . . . . . . . . . . . 19

2.5.4 Comparing fMM[(1− ε)R] with the Optimal . . . . . . . . . . . . . . 20

2.6 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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2.6.1 Comparing Our Solutions with the Optimal . . . . . . . . . . . . . . 21

2.6.2 Comparing Our Solutions with a Conceivable Baseline . . . . . . . . 24

2.7 Extension to Multiple-Unicast Networking . . . . . . . . . . . . . . . . . . . 27

2.7.1 System Model for Multiple-Unicast Networking . . . . . . . . . . . . 27

2.7.2 Existing Delay-Gap Results in the Multiple-Unicast Setting . . . . . 28

2.7.3 Extending Our Results to the Multiple-Unicast Scenario . . . . . . . 30

2.8 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Minimize Age-of-Information with Throughput Requirements 34

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


3.3 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Comparing AoI with Maximum Delay . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Minimizing AoI is NP-Hard in the Weak Sense . . . . . . . . . . . . . . . . . 44

3.5.1 MPA differ from MAA, but both are NP-Hard . . . . . . . . . . . . . 44

3.5.2 Obtaining MR in a Pseudo-Polynomial Time . . . . . . . . . . . . . . 45

3.5.3 Minimizing AoI Optimally in a Pseudo-Polynomial Time . . . . . . . 47

3.6 An Approximation Framework for Minimizing AoI . . . . . . . . . . . . . . . 48


3.7.1 Simulations on Known Topologies . . . . . . . . . . . . . . . . . . . . 50

3.7.2 Simulations on Randomly Generated Topologies . . . . . . . . . . . . 52

3.8 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Maximize Network Utility with Throughput Requirements and MaximumDelay Constraints 54

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


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4.3 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


4.4 An Algorithm PASS for Maximizing Aggregate User Utilities . . . . . . . . . 59

4.4.1 Algorithmic Structure of PASS . . . . . . . . . . . . . . . . . . . . . 59

4.4.2 PASS Solves MUDT Approximately, Violating Constraints by ConstantRatios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.3 PASS-M Solves MUDT Approximately, Violating Throughput Require-ments by Problem-Dependent Ratios . . . . . . . . . . . . . . . . . . 62

4.4.4 PASS-T Solves MUDT Approximately, Violating Delay Constraints byProblem-Dependent Ratios . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4.5 Some Other Important Maximum-Delay-Aware Problems . . . . . . . 64

4.5 Popular Network Communication Settings Sensitive Both to Throughput andto Maximum Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5.1 Minimizing Maximum Delay under Throughput Requirements . . . . 64

4.5.2 Maximizing Throughput-Based Utility under Maximum Delay Con-straints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65


4.6.1 Simulating the Problem of Minimizing Maximum Delay . . . . . . . . 66

4.6.2 Simulating the Problem of Maximizing Throughput . . . . . . . . . . 67

4.6.3 Simulating the Problem of Maximizing Network Utility . . . . . . . . 68

4.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Energy-Efficient Timely Truck Transportation for Fulfilling Multiple Tasks 70

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70


5.3 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73


5.4 It is Challenging to Solve Our Truck Transportation Problem . . . . . . . . . 77

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5.5 An Efficient Heuristic SPEED for Our Problem MEET . . . . . . . . . . . . . 78

5.5.1 A New Problem Formulation of MEET . . . . . . . . . . . . . . . . . 78

5.5.2 The Lagrangian Dual Relaxation of MEET . . . . . . . . . . . . . . . 79

5.5.3 Our Proposed Heuristic SPEED . . . . . . . . . . . . . . . . . . . . . 80


5.6.1 Fuel-Consumption-Rate Function Model . . . . . . . . . . . . . . . . 83

5.6.2 Comparing SPEED with Alternatives for Two Tasks . . . . . . . . . . 84

5.6.3 Comparing SPEED with Alternatives for Three Tasks . . . . . . . . . 85

5.6.4 Impact of Time Windows on the Fuel Consumption . . . . . . . . . . 86

5.6.5 Fuel Saving of SPEED with the Number of Tasks . . . . . . . . . . . 87

5.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Dissertation Summary and Future Work 90

6.1 Dissertation Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Bibliography 93

Appendix A Proofs of our Theorems and Lemmas 102

A.1 Proof of our Lem. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

A.2 Proof of our Thm. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

A.3 Approximate Min-Max Flow in Simulations . . . . . . . . . . . . . . . . . . 104

A.3.1 Figure Out an Upper Bound of M(fMM(R)) . . . . . . . . . . . . . . 104

A.3.2 Figure Out a Lower Bound of M(fMM(R)) . . . . . . . . . . . . . . . 105

A.3.3 Compare the Upper Bound with the Lower Bound . . . . . . . . . . . 106

A.4 Proof of Our Thm 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

A.5 Slotted Transmission Model Considers Different Kinds of Networking Delays 107

A.6 Proof of our Lem. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.7 Proof of our Lem. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.8 Proof of our Lem. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

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A.9 Proof of our Lem. 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.10 Proof of our Lem. 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.11 Proof of our Lem. 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A.12 Proof of our Lem. 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A.13 Proof of our Lem. 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A.14 Proof of our Lem. 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

A.15 Proof of our Thm. 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A.16 Proof of our Thm. 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

A.17 Proof of our Thm. 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

A.18 Proof of our Thm. 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.19 Proof of our Thm. 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

A.20 Proof of our Thm. 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

A.21 Proof of our Thm. 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

A.22 Proof of our Thm. 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

A.23 Proof of our Thm. 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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List of Figures

2.1 An illustrative example of the optimal solutions to the three network delayoptimization problems. The input network has two nodes and two parallellinks. We assume that the link delay is a constant of 1 for the dashed link,while it is a function of D(x) = xp (p > 1 and p is large) for the solid link. . 13

2.2 A network with n nodes and 2(n− 1) links. The delay of each upper dashedlink is a constant of 1, while that of each lower solid link is a function of D(x)defined in (2.19). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Simulated network GEANT [2]. . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Bi-criteria delay gaps of fSO[(1− ε)R] as compared to the optimal. . . . . . . 22

2.5 Bi-criteria delay gaps of fSF[(1− ε)R] as compared to the optimal. . . . . . . 22

2.6 Bi-criteria delay gaps of fNE[(1− ε)R] as compared to the optimal. . . . . . . 23

2.7 Bi-criteria delay gaps of fMM[(1− ε)R] as compared to the optimal. . . . . . 23

2.8 Ratios comparing the delays of fB[(1− ε)R], fSO[(1− ε)R], fSF[(1− ε)R], andfNE[(1− ε)R], respectively with that of fB(R). We set s = IS, t = IL, R = 10,and vary ε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.9 Ratios comparing the delays of fB[(1− ε)R], fSO[(1− ε)R], fSF[(1− ε)R], andfNE[(1− ε)R], respectively with that of fB(R). We set s = IS, t = IL, ε = 0.03,and vary R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.10 Ratios comparing the delays of fB[(1− ε)R], fSO[(1− ε)R], fSF[(1− ε)R], andfNE[(1−ε)R], respectively with that of fB(R). We set s = PT, t = EE, R = 15,and vary ε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.11 Ratios comparing the delays of fB[(1− ε)R], fSO[(1− ε)R], fSF[(1− ε)R], andfNE[(1−ε)R], respectively with that of fB(R). We set s = PT, t = EE, ε = 0.03,and vary R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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3.1 An illustrative example of the batch-based AoI. Suppose that sender s gen-erates a batch of two packets at each slot 3k,∀k ∈ Z. It sends the twopackets one-by-one over link (s, r) to the receiver r; the link transmissionincurs one-slot delay. As r receives all the two packets in a batch at eachslot 3k + 2,∀k ∈ Z, the batch-based AoI becomes 2 at each slot 3k + 2, i.e.,the elapsed time since the batch generation. The batch-based AoI increaseslinearly at other slot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 An example of constructing the expanded network Gexp. We assume thatd(a,r) = 1, d(s,a) = 2, MU = 5, and M = 4. . . . . . . . . . . . . . . . . . . . . 45

3.3 Two simulated network topologies in our AoI study. . . . . . . . . . . . . . . 50

3.4 Simulated AoI of two instances on networks with known topologies. . . . . . 51

3.5 Simulated AoI of 200 instances on networks with known topologies. . . . . . 52

3.6 Simulated AoI of 200 instances on networks with randomly generated topolo-gies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Simulated network of 6 Amazon EC2 datacenters [58]. . . . . . . . . . . . . . 66

4.2 Simulate the problem of minimizing maximum delay. . . . . . . . . . . . . . 67

4.3 Simulate the problem of maximizing throughput. We vary ε and assumeD1 = D2 = 150. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Simulate the problem of maximizing utility, where we set R1 = R2 = 80 andD1 = D2 = 150. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1 A truck timely fulfills multiple tasks in a national highway network. . . . . 73

5.2 Simulated truck fuel consumption rate with driving speed. . . . . . . . . . . 84

5.3 Fuel saving achieved by SPEED with the time window constraints, as com-pared to the fastest-path-based baseline. Here t∗i is defined in the equationin (5.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 Fuel saving of SPEED as compared to the PASO-based approach. We setT in

2 = t∗1 · (1 + x%) and T in3 = (t∗1 + t∗2) · (1 + x%), where t∗i is defined in the

equation in (5.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5 The fuel reduction of SPEED as compared to the fastest-path baseline, as afunction of K. All latest arrival time constraints are set to be (1+x%) ·

∑Ki=1 t

∗i . 88

A.1 A two-node two-link network. We assume that D(x) = xp (p > 1) is the delayfunction of the solid link, and D(x) = 1 is the delay function of the dashed link.103

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A.2 GEANT∗: a network reduced from GEANT. . . . . . . . . . . . . . . . . . . 105

A.3 A two-node two-link network. We assume that Ru = 10/7, Rl = 1, D = 10,the bandwidth of e1 (resp. e2) is 1 (resp. 10), and the delay of e1 (resp. e2)is 1 (resp. 11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A.4 A two-node three-link network. The bandwidth of each link is 1, the delay ofe1, e2, and e3 are 1, 6, and 7, respectively, D = 5, Rl = 1, and Ru = 5/2. . . 110

A.5 A two-node two-link network. We assume that the delay of e1 (resp. e2) is 1(resp. d), the bandwidth of e1 (resp. (e2)) is 1 (resp. 5), D = 5, Rl = 5/6,and Ru = 5/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

xv

List of Tables

2.1 Existing average delay gap (resp. maximum delay gap) of SO, NE, and MM ascompared to the respective optimal, under the single-unicast setting. fSO(R)is the solution optimal to SO, fMM(R) is the solution optimal to MM, andfNE(R) is the Nash equilibrium. All of the three solutions can support a flowrate of R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Our bi-criteria delay gaps all of which are at least near-tight. A lower bound(α′, β′) of the average delay gap (resp. maximum delay gap) of a solution de-notes that there exists an instance where this solution can support α′-fractionof the flow rate requirement, and its average delay (resp. maximum delay)must be no smaller than β′ times optimal. Our fSF[(1 − ε)R], fNE[(1 − ε)R],and fMM[(1− ε)R] are defined in Sec. 2.3.3. . . . . . . . . . . . . . . . . . . . 9

2.3 Suppose the sender is IS, the receiver is IL, and the rate requirement is 10.We give corresponding delay results of the conceivable baseline with different θ. 24

2.4 Existing delay gaps of SO, NE, and MM, under the multiple-unicast sce-nario [14,15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Bi-criteria delay gaps of our proposed solutions in the multiple-unicast setting. 32

3.1 Comparing our AoI work with existing AoI studies. . . . . . . . . . . . . . . . 35

3.2 Summary of important notations used in our AoI study. . . . . . . . . . . . . 38

4.1 Compare existing studies with our NUM work. . . . . . . . . . . . . . . . . . 55

4.2 (de, ce) of each link e ∈ E in Amazon EC2 [29, 58], where ce is capacity (inMbps) and de is delay (in ms). . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.1 Compare existing energy-efficient timely truck transportation studies with ours. 71

5.2 Summary of used notations in our truck transportation study. . . . . . . . . 74

xvi

5.3 An illustrative example of our studied truck transportation problem MEETbased on Fig. 5.1(a). We set tle = tue = 1 and cle = cue for each e ∈ E. . . . . . 77

5.4 Compare SPEED with alternatives for the instance of (1, 9, 22, 40, 65). A lowerbound of the optimal fuel consumption is 478.73 according to Thm. 26. . . . 84

5.5 Compare SPEED with alternatives for the instance of (18, 11, 4, 22, 45, 75, 105),where a lower bound of the optimal fuel consumption is 664.77 according toThm. 26. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.6 Fuel consumption reduction of SPEED as compared to the PASO-based alter-native. We set T in

2 = T in3 = (t∗1 + t∗2) · (1 + x%), where t∗i is defined in the

equation (5.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.7 Ratio of solvable MEET instances of the PASO-based alternative with thenumber of tasks. All latest arrival time constraints are set to be (1+x%)·

∑Ki=1 t

∗i . 88

xvii

Chapter 1

Introduction

1.1 Motivation and Background

Delay is known to be a critical performance metric for various real-world routing applications,e.g., for multimedia communication. Provisioning routing services with minimal End-to-End(E2E) (or sender-to-receiver, source-to-destination equivalently) delay or at least boundedE2E delay is highly vital for time-sensitive applications which nowadays become increasinglyimportant in the domains of communication and transportation. In the following we illus-trate this critical observation by three representative application scenarios, i.e., cloud videoconferencing, mobile video recognition, and freight deliveries of long-haul trucks.

First, we observe that cloud video conferencing attracts substantial attention, as recentreports say that nowadays up to 75% of innovators utilize video collaboration1, and up to 51million users attend WebEx meetings per month2. Low delay performance is a must for high-quality video conferencing solutions, considering that the International TelecommunicationUnion (ITU) [39] suggests that for video conferencing, an E2E delay within 150ms can providea service of transparent interactivity, while an E2E delay greater than 400ms is unacceptable.Therefore, it is a key problem for cloud video conferencing applications to efficiently usenetwork resources to route the conferencing traffic, obtaining acceptable delay performances.

Second, delay is also desired to be minimized or at least bounded above for the applicationof mobile video recognition. Nowadays the blending of video recognition and mobile devicesis taking place. In order to process images and make smart decisions, deep learning is ofteninvolved. Since the mobile device is resource-constrained while deep learning is resource-heavy, generally speaking, images cannot be processed locally on mobile devices frequentlyin real time. For example, the video processing rate of the deep learning model in Tensorflowis less than 1 FPS on a typical Android phone [70]. The widely-adopted solution is to process

1Cisco, http://www.cisco.com/2WebEx, https://blog.webex.com/

1

2

some images on mobile devices, while offload other images to nearby powerful edge serversfor processing. For example for the video recognition Android application developed by [70],after running it for 30 minutes, it consumes 25% battery and processes images at a 5 FPSrate if on the phone; in comparison, it consumes 15% battery and processes images at a9 FPS rate if on a server. To offload images to servers for processing, Ran et al. [70] alsosuggests that networking delay dominates (over 95%) the total delay. Therefore, routingalgorithms that provide low-delay performances are necessary for mobile video recognition.

Third, delay (also known as the travel time in the transportation field) is an important per-formance metric which is taken into account by current truck operators. Nowadays mobileapplications like uShip3 and Uber Freight4 provide a huge number of freight transportationrequests for truck operators, which are often associated with earliest/latest pickup time re-quirements and delivery time requirements. Violation of those requirements can cause a hugerevenue loss for truck operators. For example, the Texas Transportation Institute estimatesa 2011 annual truck violation of time-sensitive constraints of over 3 million hours at a costof 246 million dollars for the Tampa Bay Urbanized Area [80]. Many reasons can explainwhy violating time-sensitive constraints leads to revenue loss. For example, many companieslike Amazon nowadays have a service-level agreement with customers, where freight deliverytime is a hard constraint that cannot be violated in order to ensure customers’ satisfac-tory5. Therefore, navigation services should provide routing solutions for truck operatorswith guaranteed travel time performances.

In summary, motivated by skyrocketing interests on supporting the routing of low-delaytraffic both in the communication area and in the transportation area, in this dissertationwe study four different delay-aware multi-path routing problems. We note that multi-pathrouting is a basic paradigm of networking, for supporting applications with better Quality-of-Service (QoS). It is a natural extension of the single-path routing when streaming a largenumber of traffic over a multi-hop network while avoiding link traffic congestions. Manyexisting studies [87,88,90,91] have shown that multi-path routing can provide strictly betterQoS (smaller delay, larger throughput, etc.) as compared to the single-path routing. Weremark that for each of our four problems, it is fundamental and considers the E2E delayperformances of all traffic. Moreover, we are the first to study them. Hence our results canbe of interest for delay-sensitive routing applications, and can serve as benchmarks for futuremulti-path routing studies with delay in consideration.

3uShip, https://www.uship.com/4Uber Freight, https://freight.uber.com/5Amazon, https://www.amazon.com/

3

1.2 Literature Review

Under the multi-path setting, a sender (or source equivalently) may utilize multiple pathsto send the traffic to a receiver (or destination equivalently), where different paths can havedifferent delays, implying that different traffic unit can experience different E2E delays fol-lowing a multi-path routing solution. Many existing delay-aware multi-path routing studies,e.g. [21,71,72,78,95], consider a metric of the aggregate traffic E2E delay (also known as thetotal delay experienced by all traffic). However, consideration of such an aggregate delaymetric lacks practical significance, since as illustrated by [15], certain traffic can experi-ence an unbounded E2E delay following the multi-path routing solution which minimizes theaggregate traffic delay.

In order to provide E2E delay guarantee for all traffic, given a multi-path routing solution,the maximum delay among all utilized paths shall be taken into account. In contrast to theaggregate traffic delay which is convex with decision variables, the maximum delay whichupper bounds the E2E delays of all traffic is non-convex with decision variables. Hence almostall maximum-delay-aware multi-path routing problems are NP-hard [14,15]. It is thus highlynon-trivial to design polynomial-time algorithms to figure out multi-path routing solutionswith theoretical performance guarantee, with the maximum delay metric in consideration.

Misra et al. [64] study a multi-path routing problem of minimizing maximum delay un-der a throughput requirement, and design a Fully-Polynomial-Time Approximation Scheme(FPTAS)6. Zhang et al. [97] develop an FPTAS to minimize maximum delay further with reli-ability and differential delay constraints taken into account. Cao et al. [10] propose an FPTASto maximize throughputs under maximum delay constraints. Their FPTAS is extended by Yuet al. [96] for solving other throughput maximization problems of Internet-of-Things (IoT)applications. Note that all these studies [10, 64, 96, 97] assume a delay model where trafficexperiences a constant delay to pass a link with the streamed traffic rate upper bounded bythe capacity. We observe that all algorithms from [10,64,96,97] rely on a standard techniqueof using time-expanded networks to iteratively solve network flow problems, which is thusvery time-consuming.

As a natural extension of the constant link delay model, in the literature there exist someother studies which model the link delay as a traffic-dependent function: Correa et al. [14,15]minimize maximum delay under a throughput requirement. They design approximationalgorithms which have delay-function-dependent approximation ratios. Roughgarden [73]also minimizes maximum delay, but develop an approximation algorithm that has a network-topology-dependent approximation ratio. We observe that these problem-dependent ratiosof algorithms from [14,15,73] can be infinitely large for certain instances, implying that theirachieved delay performances can be arbitrarily bad in theory.

6In the literature FPTAS is defined as a (1 + ε)-approximation algorithm, with a time complexity that ispolynomial with all problem inputs and 1/ε, for any user-defined ε > 0.

4

Delay-constrained routing are also attracting substantial attention nowadays in the trans-portation area, e.g., for the long-haul heavy-duty truck operation. Representative studiesinclude [18, 19, 41, 59]. These studies focus on minimizing truck fuel consumption under adeadline constraint. Here the deadline constraint upper bounds the travel time for a truckto fulfill one transportation task. We note that studies [18, 19, 41, 59] only consider onetransportation task, which limits their applications. Given multiple transportation tasks ina specific order, the problem of minimizing fuel consumption becomes harder with traveltime constraints of different tasks involved, since travel time constraints can overlap in thetime domain and it requires to optimally allocate travel time budgets for individual tasks toachieve the minimal fuel consumption.

1.3 Contributions and Dissertation Structure

There are different delay-based performance metrics including average delay, maximum delay,Age of Information (AoI), and network utility: (i) average delay is the average experiencedE2E delays of all traffic, and is equal to the aggregate traffic delay over the volume oftraffic. A multi-path routing solution with low average delay can utilize resources efficientlyfrom the perspective of the whole network system; (ii) maximum delay is the maximumexperienced E2E delays of all traffic. A multi-path routing solution with low maximum delaycan provide small delay performances for all traffic; (iii) for problems where throughput canbe optimized, it is not enough to only consider delay to obtain high-quality routing solutions,since different solutions with the same delay performance can achieve different throughput.Under this setting, AoI should be taken into account, which is defined as the elapsed timesince the generation of the last received information, and hence depends both on the delayand on the throughput; (iv) in addition in general, a network user can define a utility functionwhich estimates the penalty by the experienced delay. High-quality solutions should providesmall penalties of network users.

In this dissertation, we study four fundamental multi-path routing problems, consideringthese different delay-based networking performance metrics. The four problems respectivelyare minimizing both the maximum delay and the average delay, minimizing AoI, maximizingnetwork utility, and timely energy-efficient trucking. All problems are NP-hard. Moreover,we are the first to study each of them, where we design approximation algorithms withtheoretical performance guarantee. Our studies are publicly available at [54–57], and ourspecific contributions are as follows.

B Minimize maximum delay and average delay. In Chapter 2, we consider a single-unicast network communication setting. We assume that in a multi-hop network a senderuses multiple paths to stream a flow to a receiver. We consider a delay model where thetransmission over a link incurs a delay modeled by a non-decreasing, non-negative, anddifferentiable function with the link aggregate transmission rate. We focus on two delaymetrics, i.e., the average delay and maximum delay. Existing results say that the two delay

5

metrics of a flow cannot be both within bounded-ratio gaps to the optimal. In comparison,we design three network solutions, each of which can deliver (1 − ε)-fraction of the flowwith the achieved two delay metrics both within a (1/ε)-ratio gap to the optimal, for anyuser-defined ε ∈ (0, 1). We prove that our gap is at least near-tight. Note that our gap(1/ε) is independent from the link delay function and network topology. Our solutions canbe extended to the multiple-unicast setting, where we prove that the two delay metrics ofour solutions are both within a bounded-ratio gap of (R/(Rmin · ε)) to the optimal. Here R(resp. Rmin) is the aggregate (resp. minimum) flow rate requirement of all sender-receiverpairs. We simulate a practical cloud-video conferencing application scenario. Empirically weobserve that the delay gaps are much smaller than their theoretical counterparts. Besides,empirically we observe that our solutions can achieve over 10% reduction on both delaymetrics, at a cost of losing 3% traffic, as compared to a conceivable delay-aware baselinewithout traffic loss. This study is also publicly available at [54].

B Minimize Age-of-Information. In Chapter 3, we consider a single-unicast networkcommunication setting. We assume that in a multi-hop network a sender requires to usemultiple paths to periodically send a batch of data to a receiver. We consider a delay modelwhere data experiences a constant delay to pass a link with the link aggregate data trans-mission rate upper bounded by a constant capacity. We study a multi-path routing problemof minimizing AoI subject to throughput requirements. We note that our AoI study differsfrom existing ones in that we are the first to consider the batch generation of data andmulti-path communication. Although our problem is proven to be NP-hard, we first designa pseudo-polynomial time optimal algorithm. We further show that minimizing maximumdelay and minimizing AoI are roughly equivalent, in the sense that any approximate solu-tion to the maximum delay minimization problem is also an approximate solution to theAoI minimization problem. Leveraging this understanding, we develop an approximationframework for minimizing AoI in a polynomial time complexity. Our framework can buildupon any polynomial-time α-approximation algorithm of the maximum delay minimizationproblem, e.g., the one from [64] with α = 1 + ε given any user-defined ε > 0, to constructan (α + c)-approximate solution for minimizing AoI. Here c is a constant dependent onthroughput requirements. Finally, we empirically evaluate our approaches using extensivesimulations over various network topologies. Empirically if the range of task activation pe-riod increases by 1, (i) our optimal algorithm reduces AoI by 3% than our approximationframework; however, (ii) our approximation framework reduces running time by 100% thanour optimal algorithm. This study is also publicly available at [56].

B Maximize network utility. In Chapter 4, we consider a multiple-unicast network com-munication setting. We assume in a multi-hop network there exist many network users(sender-receiver pairs). Each user requires a sender to use multiple paths to stream a flowto a receiver, incurring an utility that is a function of the experienced maximum delay orthe achieved throughput. We consider a constant link delay model that is the same asused in our AoI minimization work. We study a problem of maximizing aggregate utilitiesof all users under maximum delay constraints and throughput requirements. Our studied

6

Network Utility Maximization (NUM) problem is highly challenging, as we show that it isNP-complete either to construct an optimal solution under relaxed maximum delay con-straints or relaxed throughput requirements, or to figure out a feasible solution with all theconstraints satisfied. We then develop a polynomial-time approximation algorithm. It canobtain solutions with constant approximation ratios under realistic conditions, at the cost ofviolating constraints by up to constant-ratios. It leverages a novel technique that relates thenon-convex maximum delay to the convex average delay, and suggests a new avenue for op-timizing maximum-delay-aware network communications. Extensive simulations assuminga real-world application scenario of cloud video conferencing evaluate our algorithm. Theutility achieved by our algorithm can be up to 100% more than that achieved by conceivablebaselines, with the throughput requirements satisfied but the maximum delay constraintsviolated. We observe that the delay violation of our algorithm is acceptable for practicalvideo conferencing applications. This study is also publicly available at [57].

B Timely energy-efficient trucking. In Chapter 5, we consider a common truck operationscenario. We assume that a long-haul truck is driving in a highway network to fulfill multipletransportation tasks in order. We study a timely eco-routing problem of minimizing totalfuel consumption under task pickup and delivery time window constraints. Our designspaces include path planning, speed planning, and task execution times optimization. Taskexecution times optimization is a new challenging design space for saving fuel in our multi-task setting, and it differentiates our study from existing ones under the single-task setting.We prove that the problem is NP-hard. Moreover, we observe that it is a non-convex puzzleeven to optimize task execution times by itself. We then develop a fast and efficient heuristic.Our heuristic can generate an optimal solution under derived conditions, and we furthercharacterize an optimality gap for it when the conditions are not met. We simulate a heavy-duty truck driving across the US national highway system, and empirically observe that thefuel consumption achieved by our heuristic can be up to 22% less than that achieved by thefastest-/shortest- path baselines, and up to 10% less than that achieved by a conceivablealternative extended from the state-of-the-art single-task algorithm. Besides, we observethat the fuel saving of our heuristic as compared to the baselines is robust to the number oftasks. This study is also publicly available at [55].

Chapter 2

Simultaneously Minimize MaximumDelay and Average Delay withThroughput Requirements

2.1 Introduction

We consider a network communication setting in a multi-hop network, where a sender re-quires to use multiple paths to stream a flow at a fixed rate to a receiver. We consider adelay model where the experienced delay of passing a link is assumed to be a differentiable,non-decreasing, and non-negative function of the link aggregate transmission rate. We studythe minimization of two E2E delay metrics, i.e., the experienced average sender-to-receiverdelay and the experienced maximum sender-to-receiver delay, from the perspective of threefundamental network delay optimization problems. The three problems are (i) problem MM,i.e., minimizing maximum delay; (ii) problem SO, i.e., minimizing average delay; and (iii)problem NE, i.e., finding the Nash equilibrium. For each of the three problems, we try to finda solution such that it can obtain a close-to-optimal maximum delay and a close-to-optimalaverage delay.

It is known that E2E delay mainly includes the propagation delay, queuing delay, and pro-cessing delay. We note that they all can be modeled as functions of flow rate in general. Forexample according to queuing theory, queuing delay can be estimated as follows [27]

D(x) =

1c−x if c > x,

+∞ otherwise,(2.1)

where x is the assigned traffic and c is the capacity. The function in (2.1) does not allowthe assigned rate to exceed the capacity. Kleinrock [46] proposes another delay formula thatconsiders the case where the assigned rate exceeds the capacity. In this chapter, we do not

7

8

Table 2.1: Existing average delay gap (resp. maximum delay gap) of SO, NE, and MM ascompared to the respective optimal, under the single-unicast setting. fSO(R) is the solutionoptimal to SO, fMM(R) is the solution optimal to MM, and fNE(R) is the Nash equilibrium.All of the three solutions can support a flow rate of R.

Average delay gap Maximum delay gapfSO(R) 1 γ(L) [14, 15]fNE(R) σ(L) [14, 15]; 1.33∗ [74]; 1.63∗, 1.90∗ [72] σ(L) [14, 15]; (|V | − 1) [73]fMM(R) σ(L) [14, 15] 1Note. Unless specified, gaps are independent to network topologies and link delay functions.

∗1.33 holds for linear delay functions, 1.63 for quadratic delay functions, and 1.90 for cubic delay functions.

restrict ourselves to specific link delay functions. Instead, we assume arbitrary link delayfunctions that are differentiable, non-decreasing, and non-negative. Note that similar linkdelay models have been used by many earlier works, e.g., [7, 14, 15,73].

The two delay metrics are both practically important for time-sensitive networking applica-tions, and are optimized by the three network delay optimization problems differently. Letus consider the cloud video conferencing with multiple sessions as an example. The optimalsolution to MM provides a good performance for every session, since minimizing the max-imum E2E delay implies that all sessions can experience an upper bounded minimal delay.The optimal solution to SO minimizes average E2E delay and gives the most efficient routingsolution of utilizing network resources in terms of the cloud. NE gives the best fair solutionwhere conferencing sessions with the same sender and receiver experience the same delay.

For delay-critical applications, an ideal traffic routing approach should simultaneously opti-mize the two delay metrics, so as to benefit the whole network as well as individual users.In particular, (i) it is desirable to optimize average delay to efficiently utilize network re-sources; (ii) in order to provide a satisfactory delay guarantee for each user, the maximumdelay should be at least bounded above by a tolerant value (e.g., 400ms for cloud videoconferencing [39]).

We summarize existing results on optimizing the two delay metrics in Tab. 2.1 and givedetails later in Sec. 2.2. Overall we observe that Correa et al. [14, 15] give the state-of-the-art, according to which for general network topologies and arbitrary link delay functions, (i)the average delay of the optimal solution to MM, the average delay of the optimal solutionto NE, and the maximum delay of the optimal solution to NE, are all bounded above by aratio of σ(L) as compared to the respective optimal, where σ(L) is defined as

σ(L) =

(1− sup

D(·)∈L,0≤x≤R

x · (D(R)−D(x))

R · D(R)

), (2.2)

where L is the set of link delay functions and R is the flow rate requirement. (ii) Themaximum delay of the optimal solution to SO is bounded above by a ratio of γ(L) as

9

Table 2.2: Our bi-criteria delay gaps all of which are at least near-tight. A lower bound(α′, β′) of the average delay gap (resp. maximum delay gap) of a solution denotes that thereexists an instance where this solution can support α′-fraction of the flow rate requirement,and its average delay (resp. maximum delay) must be no smaller than β′ times optimal. OurfSF[(1− ε)R], fNE[(1− ε)R], and fMM[(1− ε)R] are defined in Sec. 2.3.3.

Bi-criteria average delay gap Bi-criteria maximum delay gapBi-criteria gap Its lower bound Bi-criteria gap Its lower bound

fSF[(1− ε)R] (1− ε, 1) (1− ε, 1) (1− ε, 1/ε) (1− ε, d1/εe − 1)fNE[(1− ε)R] (1− ε, 1/ε) (1− ε, 1/ε) (1− ε, 1/ε) (1− ε, d1/εe − 1)fMM[(1− ε)R] (1− ε, 1/ε) (1− ε, 1/ε) (1− ε, 1) (1− ε, 1)

Note. ε can be any user-defined value that is greater than 0 and less than 1.

compared to the optimal. Here γ(L) is the smallest number meeting the following

D(x) + x · D′(x) ≤ γ(L) · D(x), ∀D(·) ∈ L,∀x ∈ [0, R]. (2.3)

In addition, (iii) all of their proposed delay gaps are proven to be tight in the single-unicastsetting. Because their problem-dependent gaps can be infinitely large for certain delayfunctions (e.g., for the queuing delay function in (2.1)), and are tight [14, 15], we observethat no optimal solutions to MM, SO, or NE can minimize the two delay metrics both withinbounded-ratio gaps as compared to the optimal.

In comparison, we propose a new approach to minimize the two delay metrics both withinconstant-ratio gaps to the optimal, at a cost of only supporting a controllable fraction of theflow rate requirement. For easier reference, we use an (α, β) (α ∈ (0, 1)) bi-criteria averagedelay gap (resp. bi-criteria maximum delay gap) of a solution to refer to that the averagedelay (resp. maximum delay) of this solution must be within β times optimal, at a costof only supporting α-fraction of the flow rate requirement. Our results are summarized inTab. 2.2, according to which our proposed bi-criteria delay gaps are all at least near-tight,and are all constants independent to network topology and link delay function.

Our approach is to either directly solve the flow problems subject to a smaller flow raterequirement (Theorems 5 and 6 for NE, Theorems 7 and 8 for MM), or sacrifice a fraction ofrates from the solutions to flow problems which are subject to a full flow rate requirement(Theorems 3 and 4 for SF, i.e., Sacrificing Flow from SO). We evaluate the empirical perfor-mance of our solutions by extensive simulations using a real-world continent-scale networktopology. We observe that the empirical delay gaps are much smaller than their theoreticalcounterparts. We also observe that our solutions can achieve over 10% reduction simultane-ously on the maximum delay and on the average delay, only at a cost of losing 3% traffic, ascompared to a conceivable delay-aware baseline without traffic loss.

10


Studies on NE many studies on NE exist to characterize the gap of the average delay ofa Nash equilibrium as compared to the optimal. First, for general network topologies butspecial link delay functions, the gap is bounded by constant-ratios. Specifically, it is 1.33 forlinear delay functions [74], 1.63 for quadratic delay functions [72], and 1.90 for cubic delayfunctions [72]. Under the linear delay function model, Christodoulou et al. [13] generalizethe result of [74] that holds for a Nash equilibrium, and show that the gap is 4 ·(1+ε)/(3−ε)for any 0 ≤ ε ≤ 1, and is (1 + ε)2 for any ε > 1, in terms of an (1 + ε)-approximate Nashequilibrium. Next, for general network topologies and arbitrary link delay functions, the gapis bounded by a problem-dependent ratio of σ(L) [14, 15] that is defined in Equation (2.2).Basu et al. [7] generalize the result of [14, 15] and give the average delay gap of an (1 + ε)-approximate Nash equilibrium.

There also exist many studies on NE to characterize the gap of the maximum delay of a Nashequilibrium as compared to the optimal. To our best knowledge, it is first introduced by [47],followed by subsequent studies including [16, 63], all for a specific two-node network withparallel links, assuming linear link delay functions. The gap is proven to be minor undercertain conditions, e.g., it is Ω(log |E|/ log log |E|) if links are identical [47], where |E| is thenumber of links. Czumaj et al. [16] also show that an equilibrium exists to minimize themaximum delay, if there are infinitely many players. Note that our study is different fromthose of [16,47,63] in two aspects: (i) for the network game model, we assume infinitely manyplayers, while existing studies [16, 47, 63] can work with a finite number of players; (ii) forthe network communication model, existing studies [16, 47, 63] focus on a specific two-nodenetwork with parallel links assuming linear link delay functions, while we consider generalnetwork topologies and arbitrary link delay functions. Under the network game model ofinfinitely many players for general network topologies and arbitrary link delay functions,Weitz [93] observes that any average delay gap of NE is also a maximum delay gap of NEin the single-unicast setting. Roughgarden [73] proves (|V | − 1) to be the maximum delaygap of NE, where |V | is the number of nodes. Note that one can formulate NE as a convexprogram, and hence solve it in a polynomial time under our system model [15].

Studies on SO according to the definition of SO, its optimal solution minimizes averagedelay. However, there exist few studies which characterize its maximum delay gap comparedto the optimal, except for [14, 15], to our best knowledge. Correa et al. [14, 15] prove thatthe maximum delay of the optimal solution to SO is bounded by a ratio of γ(L) as comparedto the minimal maximum delay, where γ(L) is defined by the inequalities in (2.3). SO canbe formulated as a convex program, if link delay functions are convex [15].

Studies on MM existing studies on MM mainly focus on designing efficient algorithms,since MM is NP-hard [15]. For example, Devetak et al. [22] develop heuristic approaches

11

without performance guarantee. For a special network where the link delay function is aconstant within a capacity, and is unbounded otherwise, Misra et al. [64] design an FPTAS,and Zhang et al. [97] develop another FPTAS further with reliability and differential delayconstraints involved. To our best knowledge, studies [14, 15] are the only ones which (i)design approximation algorithms for MM considering arbitrary link delay functions, and (ii)characterize the average delay gap of the optimal solution to MM as compared to the minimalaverage delay. They prove that the gap is bounded by σ(L) defined in Equation (2.2).

Overall, for general network topologies and arbitrary link delay functions, solving NE, SO,and MM all can minimize the two delay metrics with problem-dependent approximationratios (σ(L) or γ(L)). Considering that those ratios can be infinitely large for certain delayfunctions (e.g., for the queuing delay function in (2.1)), and are tight [14,15], existing studiessay that the two delay metrics of optimal solutions to NE, SO, or MM cannot be both withinbounded-ratio gaps as compared to the optimal. In comparison, in this chapter we developmultiple solutions each of which can minimize the two delay metrics both within bi-criteriaconstant-ratio gaps as compared to the optimal.

2.3 Preliminary

2.3.1 Two Delay Metrics

We use a directed graph G , (V,E) to model a multi-hop network. We assume |E| (resp.|V |) to be the number of links (resp. nodes). Data experiences a delay of De(xe) to pass alink e ∈ E. We assume De(xe) to be non-decreasing, non-negative, and differentiable withthe aggregate link rate xe. In the network G a sender s ∈ V requires to use multiple pathsto stream a flow at a rate of R > 0 to a receiver t ∈ V \s.

Suppose P is the set of simple sender-to-receiver paths. We define a network flow f as therates allocated to each p ∈ P , i.e., f , xp : xp ≥ 0, p ∈ P. We define |f | as the total ratesent from s to t following f , i.e.,

|f | ,∑p∈P

xp. (2.4)

We define xe as the aggregate rate allocated to e by the solution f , i.e.,

xe ,∑

p∈P :e∈p

xp, (2.5)

and the delay of the path p under flow f is

dp(f) ,∑e∈p

De(xe). (2.6)

12

Two critical delay metrics of a solution f is the average delay and the maximum delay. Wedefine M(f) as the maximum delay of f , which is the largest path delay among all pathsthat are assigned positive rates1,

M(f) , maxp∈P :xp>0

dp(f). (2.7)

We define T (f) as the total delay of f , which is the aggregate delay experienced by all flowunits,

T (f) ,∑p∈P

dp(f) · xp =∑e∈E

De(xe) · xe. (2.8)

With M(f) and T (f), we can easily define A(f) as the average delay of f

A(f) , T (f)/|f |. (2.9)

2.3.2 Three Delay-Aware Network Flow Problems

MM: the Maximum delay Minimization problem is to minimize maximum delay subject toa flow rate requirement,

MM : minfM(f) s.t. |f | = R.

We use fMM(R) to denote an arbitrary optimal solution to MM under rate requirement R.We also call fMM(R) a min-max flow. The minimal maximum delay that can be achieved bya flow supporting a rate of R is denoted asM∗(R). Clearly, we haveM∗(R) =M(fMM(R)).

SO: the System Optimization problem is to minimize average delay subject to a flow raterequirement,

SO : minfA(f) s.t. |f | = R.

We define fSO(R) as an arbitrary optimal solution to SO under rate requirement R. Wedefine A∗(R) as the minimal average delay that can be achieved by a flow supporting a rateof R. To facilitate later analysis, we also define the minimal total delay as T ∗(R). It isclear that we have A∗(R) = A(fSO(R)), and T ∗(R) = R · A∗(R) = T (fSO(R)). We also callfSO(R) a system-optimal flow.

NE: to find a Nash Equilibrium flow subject to a flow rate requirement. We consider a non-cooperative network game of infinitely many players. Each player can control an arbitrarilysmall amount of flow rate and acts selfishly to route its rate from the sender to the receiver tominimize the experienced delay. A Nash equilibrium of this game has the following definition.

Definition 1 ( [14,15,72–74]). A flow f is a Nash equilibrium flow if for any pair of pathsp1, p2 ∈ P with xp1 > 0, we have dp1(f) ≤ dp2(f).

1We call a path a flow-carrying path, if it has a positive rate.

13

s tx ≤1

1-x

s t(p+1)-1/p

1-(p+1)-1/p

s t1

0

Min-max flow System-optimal flow

Nash flow

Source: sReceiver: tR=1

Figure 2.1: An illustrative example of the optimal solutions to the three network delayoptimization problems. The input network has two nodes and two parallel links. We assumethat the link delay is a constant of 1 for the dashed link, while it is a function of D(x) = xp

(p > 1 and p is large) for the solid link.

Problem NE has the following definition.

NE : find a Nash equilibrium flow f subject to |f | = R.

We define fNE(R) as an arbitrary Nash flow under rate requirement R. By its definition, allflow-carrying paths of a Nash flow share the same path delay, i.e., we have M[fNE(R)] =A[fNE(R)] = T [fNE(R)]/R. As proved in [74, Lemma 2.6], with our link delay model, a Nashequilibrium flow must exist, and all the Nash equilibrium flows share the same average delay.

We illustrate the three different delay-aware network flow solutions using an example inFig. 2.1. In this illustrative example, any feasible flow from s to t is a min-max flow; thesystem-optimal flow streams (p+ 1)−1/p rate to the solid link and the remaining rate to thedashed one; and the Nash flow uses the lower link to send all the flow rate R = 1.

We remark again that existing results suggest that none of the optimal solutions to thosethree problems can minimize the two delay metrics both within bounded-ratio gaps to theoptimal, for general network topologies and arbitrary link delay functions. Instead, we posean optimistic note that the two delay metrics can be optimized to be simultaneously withinbi-criteria constant-ratio gaps to the optimal. Our approaches are defined in the followingsection.

2.3.3 Minimizing the Two Delay Metrics by Sacrificing Flow Rate

First, we can define the following three solutions by directly solving the aforementionednetwork flow problems subject to a smaller rate of (1− ε)R, given an ε ∈ (0, 1):

14

• fMM[(1−ε)R]: directly solve the problem MM under a flow rate requirement of (1−ε)Rand figure out the optimal solution;

• fSO[(1− ε)R]: directly solve the problem SO under a flow rate requirement of (1− ε)Rand figure out the optimal solution;

• fNE[(1− ε)R]: directly solve the problem NE under a flow rate requirement of (1− ε)Rand figure out the optimal solution;

However, we are only able to prove bi-criteria constant-ratio gaps for fMM[(1 − ε)R] andfNE[(1 − ε)R], but not for fSO[(1 − ε)R]. This motivates us to further develop anotheralgorithm (Algorithm 1) to get a solution with sacrificed flow rate based on the problemSO, such that the achieved two delay metrics can be proven to be both within bi-criteriaconstant-ratio gaps to the optimal:

• fSF[(1− ε)R]: the solution by Sacrificing ε-fraction of the Flow rate from fSO(R) usingAlgorithm 1.

The algorithmic details of obtaining fSF[(1− ε)R] are given in the following section.

2.4 A Flow-Sacrificing Algorithm to SO

We now design an algorithm which gives a flow solution fSF[(1−ε)R] that sacrifices ε-fractionof the flow rate from fSO(R). fSF[(1−ε)R] can minimize the two delay metrics simultaneouslywithin bi-criteria constant-ratio gaps to the optimal.

2.4.1 Obtaining fSF[(1− ε)R] in a Polynomial Time

Algorithm 1 gives details of obtaining fSF[(1− ε)R]: we first obtain a path-defined system-optimal flow fSO(R) (Line 4). Then we iteratively delete (ε · R) rate from the slowestflow-carrying paths of fSO(R) (Lines 6–13). Finally the remaining flow is fSF[(1− ε)R].

Note that if link delay functions are convex, a path-defined system-optimal flow can beobtained in a polynomial time [15]: we can formulate SO by a convex program using anedge-defined flow formulation [8], where the size of the program is polynomial. Its optimalsolution is an edge-defined system-optimal flow and can be converted to a path-defined flowby a polynomial-time flow decomposition [24]. The flow decomposition outputs at most|E| flow-carrying paths [24], implying that the maximum number of iterations of the loop(Lines 6–13) is |E|.

15

Algorithm 1 Obtain fSF[(1− ε)R]

1: input: G = (V,E), s, t, R, ε ∈ (0, 1)2: output: fSF[(1− ε)R]3: procedure4: fSO(R) is a path-defined optimal solution of the problem SO with inputs (G,R, s, t)5: xdelete = ε ·R6: while xdelete > 0 do7: Find the slowest flow-carrying path pl ∈ P8: if xpl > xdelete then9: xpl = xpl − xdelete

10: xdelete = 011: else12: xdelete = xdelete − xpl13: xpl = 014: The remaining flow is our solution fSF[(1− ε)R]15: return fSF[(1− ε)R]

2.4.2 Critical Delay Properties of fSF[(1− ε)R]

We observe that the proposed fSF[(1− ε)R] has the following critical delay properties.

Lemma 1. Suppose fSF((1− ε)R) is the solution given by Algorithm 1. The following musthold

A[fSF((1− ε)R)] ≤ A∗(R), (2.10)

M[fSF((1− ε)R)] ≤ A∗(R)/ε. (2.11)

Proof. Refer to our Appendix A.1.

According to Lem. 1, we know that (i) the average delay of fSF[(1 − ε)R] must be within abi-criteria gap of (1− ε, 1) to the optimal; (ii) the maximum delay of fSF[(1− ε)R] must bewithin a bi-criteria gap of (1− ε, 1/ε) to the optimal, due to that we have A∗(R) ≤M∗(R).In Theorems 3 and 4 of the following section, we further prove that both gaps are at leastnear-tight. Overall, we observe that Algorithm 1 generates a good traffic routing solution fordelay-sensitive applications, since the achieved two delay metrics are both close-to-optimal.

2.5 Constant Bi-Criteria Delay Gaps

In this section, we give optimality gaps of the two delay metrics, in terms of our proposedflow solutions where we sacrifice ε-fraction of flow rate given any ε ∈ (0, 1). We summarize

16

our results in Tab. 2.2. Note that all solutions in this section can support (1 − ε)-portionof the flow rate requirement, which is straightforward based on their respective definitionsintroduced in Sec. 2.3.3.

2.5.1 Comparing fSO[(1− ε)R] with the Optimal

Comparing the delay of fSO[(1−ε)R] that supports a sacrificed flow rate of (1−ε)·R with theoptimal delay performance subject to a full rate requirement of R, we show that its averagedelay must be no worse than optimal in Thm. 1. However, comparing its maximum delaywith the optimal, the gap is problem-dependent as introduced in Thm. 2.

Theorem 1. For any given ε ∈ (0, 1), the following holds for A(fSO[(1− ε)R])

A(fSO[(1− ε)R]) ≤ A∗(R). (2.12)

Further, an instance must exist such that given any ε ∈ (0, 1)

A(fSO[(1− ε)R]) = A∗(R). (2.13)

Proof. Given a rate requirement of (1 − ε) · R, since fSO[(1 − ε)R] is optimal to SO andfSF[(1− ε)R] is feasible to SO, we have

A(fSO[(1− ε)R]) ≤ A(fSF[(1− ε)R]).

Then based on the inequality in (2.10), we can prove the inequality in (2.12). Given anyε ∈ (0, 1), consider a simple network with two nodes and one link that has a constant linkdelay. It is straightforward to verify that the equality in (2.13) holds in this example.

Theorem 2. The following holds for M(fSO[(1− ε)R])

M(fSO[(1− ε)R]) ≤ γ(L) · M∗(R), (2.14)

where γ(L) is introduced by [14, 15], and is defined by the inequalities in (2.3).

Proof. Because link delay functions are non-decreasing, we have

M∗([1− ε]R) ≤ M∗(R).

As introduced in our Tab. 2.1 and proven by existing studies [14,15], it holds that

M(fSO[1− ε]R) ≤ γ(L) · M∗([1− ε]R).

Therefore, the inequality in (2.14) holds.

17

We remark that the problem-dependent gap γ(L) can be infinite in theory. For example,considering the queuing delay function in (2.1), following the definition of γ(L) and assuminga rate requirement of R < c, we have

γ · (c− x) ≥ c, ∀ x ∈ [0, R],

implying an infinitely large γ since R can be arbitrarily close to c.

Overall, we observe that directly solving SO under a smaller rate requirement does notimprove the theoretical gap on the maximum delay. Hence, the solution fSO[(1− ε)R] couldstill lead to unsatisfying users experience due to large delays, even when users are willing tolose some traffic.

2.5.2 Comparing fSF[(1− ε)R] with the Optimal

Comparing the delay of fSF[(1−ε)R] that supports a sacrificed flow rate of (1−ε) ·R with theoptimal delay performance subject to a full rate requirement of R, we prove that its averagedelay is no worse than optimal in Thm. 3, and its maximum delay is within a constant-ratiogap to the optimal in Thm. 4.

Theorem 3. For any given ε ∈ (0, 1), the following holds for A(fSF[(1− ε)R])

A(fSF[(1− ε)R]) ≤ A∗(R). (2.15)

Further, an instance must exit such that given any ε ∈ (0, 1)

A(fSF[(1− ε)R]) = A∗(R). (2.16)

Proof. Inequality (2.15) holds due to Lem. 1, and (2.16) holds in the same example as thatintroduced in Thm. 1.

Theorem 4. For any given ε ∈ (0, 1), the following holds for M(fSF[(1− ε)R])

M(fSF[(1− ε)R]) ≤ (1/ε) · M∗(R). (2.17)


M(fSF[(1− ε)R]) ≥ (d1/εe − 1) · M∗(R). (2.18)

Proof. Based on the inequality in (4.4), we have

M[fSF((1− ε)R)] ≤ A∗(R)/ε ≤ M∗(R)/ε.

We prove the inequality in (2.18) using Fig. 2.2.

18 Figure 2.2: A network with n nodes and 2(n− 1) links. The delay of each upper dashed linkis a constant of 1, while that of each lower solid link is a function of D(x) defined in (2.19).

Because we have 1/ε > d1/εe − 1 given any ε ∈ (0, 1), there must exist an α > 1 with1/(αε) = d1/εe − 1. Suppose n = d1/εe = 1/(αε) + 1 ≥ 2. Now consider an instanceintroduced by Fig. 2.2, where we assume that R = 1, s = a1, and t = an. The delay ofeach dashed link is a constant of 1, while that of each solid link is a function defined in thefollowing

D(x) =

0, if 0 ≤ x ≤ n−2n−1

,[(x− n−2

n−1

)/(

1−1/αn−1

)]2

, if x > n−2n−1

.(2.19)

It is clear that D(n−2n−1

) = 0, D(n−1−1/αn−1

) = 1, and D(R) = D(1) > 1.

In this instance we have M∗(R) = 1, because we can construct a solution which has amaximum delay of 1 and a flow rate of 1, as well as thatM∗(R) ≥ 1 holds straightforwardlyin this example. We note that there are (n− 1) different s− t paths containing exactly onedashed link. We assign a flow rate of 1/(n − 1) to each of these (n − 1) paths. Each solidlink thus has a rate of n−2

n−1, leading to a link delay of D(n−2

n−1) = 0. Therefore, the path delay

of each of these (n− 1) flow-carrying paths is 1, implying that the achieved maximum delayis also 1.

Next we show that M(fSF[(1 − ε)R]) = n − 1 = d1/εe − 1. According to [74, Corollary2.5], we can get fSO(R) by figuring out fNE(R) with new link delay functions De(xe) =De(xe) + xe · D′e(xe). In our example, the new link delay function for dashed links is 1. Forsolid links it is D(x) = D(x) + x · D′(x), and we observe that D(x) is non-decreasing for

x ≥ 0 and strictly increasing for x ∈ [n−2n−1

, n−1−1/αn−1

]. Also we have

D(n− 2

n− 1

)= 0, D

(n− 1− 1/α

n− 1

)> 1.

Hence we can define λ ∈ (n−2n−1

, n−1−1/αn−1

) such that D(λ) = 1. Now we know that in fSO(R),each solid link is assigned a rate of λ and each dashed link is assigned a rate of (1− λ).

The path-defined solution of fSO(R) can be: a rate of (1 − λ) is assigned to the path p1

containing all dashed links, leading to a delay of n − 1 for p1; and a rate of λ is assignedto the path p2 containing all solid links, leading to a delay of (n − 1)D(λ) < n − 1 for p2.Following Algorithm 1, we learn that all ε ·R rate will be deleted from p1. Because p1 has astrictly positive rate after deleting ε ·R rate, we have

M(fSF[(1− ε)R]) = n− 1.

19

Therefore, the following holds in our example

M(fSF[(1− ε)R])

M∗(R)= n− 1 = d1/εe − 1,

which proves the inequality in (2.18).

fSF[(1−ε)R] differs from fSO[(1−ε)R] in that it minimizes the two delay metrics both withinconstant-ratio gaps to the optimal. Note that as discussed in Section 2.4.1, fSF[(1 − ε)R],can be obtained quickly in a polynomial time if link delay functions are convex.

2.5.3 Comparing fNE[(1− ε)R] with the Optimal

Compared with SO, we can solve NE in a polynomial time even if the link delay functionsare non-convex [15]. Comparing the delay of fNE[(1 − ε)R] that supports a sacrificed flowrate of (1 − ε) · R with the optimal delay performance under a full rate requirement of R,we prove that the two delay metrics must be both bounded by constant-ratio gaps to theoptimal, in Thm. 5 and 6.

Theorem 5. For any given ε ∈ (0, 1), the following holds for A(fNE[(1− ε)R])

A(fNE[(1− ε)R]) ≤ (1/ε) · A∗(R). (2.20)


A(fNE[(1− ε)R]) = (1/ε) · A∗(R). (2.21)

Proof. It is an adaptation of [74, Thm. 3.2]. Details refer to our Appendix A.2.

Theorem 6. For any given ε ∈ (0, 1), the following holds for M(fNE[(1− ε)R])

M(fNE[(1− ε)R]) ≤ (1/ε) · M∗(R). (2.22)


M(fNE[(1− ε)R]) ≥ (d1/εe − 1) · M∗(R). (2.23)

Proof. Leveraging Thm. 5 and the definition of a Nash flow, we have

M(fNE[(1− ε)R]) = A(fNE[(1− ε)R]) ≤ A∗(R)

ε≤ A(fMM(R))

ε≤ M

∗(R)

ε.

Inequality (2.23) holds for the same example of Fig. 2.2 that is introduced in Thm. 4.

We remark that although fNE[(1 − ε)R] obtains a close-to-optimal maximum delay and aclose-to-optimal average delay, by comparing Thm. 3 with Thm. 5, theoretically speaking,its average delay gap is worse than that of fSF[(1− ε)R].

20

2.5.4 Comparing fMM[(1− ε)R] with the Optimal

Comparing the delay of fMM[(1− ε)R] that supports a sacrificed flow rate of (1− ε) ·R withthe optimal delay performance under a full rate requirement of R, we prove that its averagedelay is bounded by a constant ratio to the optimal in Thm. 7 and its maximum delay is noworse than optimal in Thm. 8.

Theorem 7. For any given ε ∈ (0, 1), the following holds for A(fMM[(1− ε)R])

A(fMM[(1− ε)R]) ≤ (1/ε) · A∗(R). (2.24)


A(fMM[(1− ε)R]) = (1/ε) · A∗(R). (2.25)

Proof. Leveraging Thm. 5 and the definition of a Nash flow, we have

A(fMM[(1− ε)R]) ≤M(fMM[(1− ε)R]) ≤M(fNE[(1− ε)R]) = A(fNE[(1− ε)R]) ≤ A∗(R)/ε.

Following a similar proof as that used to prove the equality in (2.21), we can prove theequality in (2.25). The intuition is that fNE[(1 − ε)R] in the example used to prove theequality in (2.21) also minimizes the maximum delay subject to a flow rate of (1− ε) ·R.

According to our Thm. 7, although different path-defined min-max flows supporting a flowrate of (1 − ε) · R can have different average delays, those average delays are always upperbounded by a constant ratio of (1/ε) to the optimal one which is under a full rate requirementof R.

Theorem 8. For any given ε ∈ (0, 1), the following holds for M(fMM[(1− ε)R])

M(fMM[(1− ε)R]) ≤ M∗(R). (2.26)


M(fMM[(1− ε)R]) = M∗(R). (2.27)

Proof. The non-decreasing property of link delay functions implies the inequality in (2.26).The equality in (2.27) holds in the same example introduced in Thm. 1.

For general network topologies and arbitrary link delay functions, existing results pessimisti-cally suggest that a flow cannot minimize the two delay metrics both within bounded-ratiogaps to the optimal. As a comparison, we optimistically suggest that the two delay met-rics are “largely” compatible, as we propose multiple flow solutions, i.e., fSF[(1 − ε)R],fNE[(1− ε)R], and fMM[(1− ε)R], all of which can obtain a maximum delay and an averagedelay simultaneously within a bi-criteria constant-ratio gap of (1− ε, 1/ε) to the optimal, forany ε ∈ (0, 1).

21

Figure 2.3: Simulated network GEANT [2].

2.6 Performance Evaluation

In order to empirically evaluate our theoretical results, we simulate a real-world continent-scale network GEANT (see our Fig. 2.3). There are 45 nodes and 57 undirected links [2].We assume a queuing delay function in (2.1) for each link delay, where the capacity ce ∈5, 10, 20, 30, 100 (Gbps) of a link e ∈ E is set according to practical evaluations [2]. Eachundirected link is treated as two directed links with opposite directions but identical ca-pacities, same to the setting in [27]. We run simulations on a laptop with 8 GB memoryand a Core i5 (2.40 GHz) processor. We use CVX [26] in Matlab to solve the convex pro-grams of the edge-defined system-optimal flow and Nash flow. All the other experiments areimplemented in C++.

2.6.1 Comparing Our Solutions with the Optimal

Suppose 1000 sessions are streaming video conferencing traffic from the node Iceland (i.e.,sender is IS) to the node Israel (i.e., receiver is IL), where the traffic demand is 10Mbps foreach session. Thus we assume a total flow rate requirement of R = 10Gbps.

Note that we cannot figure out fMM(R) in our simulation since it is NP-hard to minimizemaximum delay and the network is dense (over 30 different paths exist in GEANT fromIS to IL). Instead we find another feasible flow fMM(R) to approximate fMM(R) efficiently,

22

(a) Maximum delay (b) Average delay

Figure 2.4: Bi-criteria delay gaps of fSO[(1− ε)R] as compared to the optimal.

0.2 0.4 0.6 0.80

5

10

Flow Rate Ratio (1−ε)

Ma

xim

um

De

lay R

atio

Empirical(1 − ǫ, 1/ǫ)(1 − ǫ, ⌈1/ǫ⌉− 1)

(a) Maximum delay

0.2 0.4 0.6 0.80

0.5

1

1.5

2


Ave

rag

e D

ela

y R

atio

Empirical(1 − ǫ, 1)

(b) Average delay

Figure 2.5: Bi-criteria delay gaps of fSF[(1− ε)R] as compared to the optimal.

since M(fMM(R)) is no worse than 1.73% more of M(fMM(R)) in average among extensivesimulation instances. Details of obtaining fMM(R) refer to Appendix A.3.

Comparing the maximum delay of our solutions with the optimal. First withoutsacrificing flow, the ratio comparing the maximum delay of fSO(R) to that of fMM(R) is1.128, and the ratio comparing the maximum delay of fNE(R) to that of fMM(R) is 1.017.We increase ε from 0.01 to 0.99 with a step of 0.01. Fig. 2.4(a) presents the ratio comparingthe maximum delay of fSO[(1− ε)R] to that of fMM(R). In the figure we do not present thetheoretical bi-criteria maximum delay gap (1 − ε, γ(L)), because as discussed in Sec. 2.5.1,γ(L) is unbounded for the queuing delay function in (2.1). As shown by the figure, differentfrom the unbounded theoretical maximum delay gap, the empirical maximum delay gap offSO[(1− ε)R] as compared to the optimal is very small. In addition, it is clear that a largersacrifice of the flow rate requirement results in a smaller maximum delay performance.

Fig. 2.5(a) gives the ratio comparing the maximum delay of fSF[(1− ε)R] to that of fMM(R).Similar to Fig. 2.4(a), the empirical maximum delay gap of fSF[(1− ε)R] as compared to the

23

0.2 0.4 0.6 0.80

5

10


Ma

xim

um

De

lay R

atio

Empirical(1 − ǫ, 1/ǫ)(1 − ǫ, ⌈1/ǫ⌉− 1)

(a) Maximum delay

0.2 0.4 0.6 0.80

5

10


Ave

rag

e D

ela

y R

atio

Empirical(1 − ǫ, 1/ǫ)

(b) Average delay

Figure 2.6: Bi-criteria delay gaps of fNE[(1− ε)R] as compared to the optimal.

0.2 0.4 0.6 0.80

0.5

1

1.5

2


Ma

xim

um

De

lay R

atio

Empirical(1 − ǫ, 1)

(a) Maximum delay

0.2 0.4 0.6 0.80

5

10


Ave

rag

e D

ela

y R

atio

Empirical(1 − ǫ, 1/ǫ)

(b) Average delay

Figure 2.7: Bi-criteria delay gaps of fMM[(1− ε)R] as compared to the optimal.

optimal is much smaller than the theoretical counterpart. For an acceptable small trafficloss for video conferencing, e.g., ε = 0.03, the maximum delay gap of fSF[(1− ε)R] is 1.007,which is better than 1.128 that is the maximum delay gap without traffic loss.

Fig. 2.6(a) illustrates the maximum delay gap of fNE[(1− ε)R] as compared to the optimal.For ε = 0.03, the gap with sacrificed flow rate is 0.953 which is better than 1.017 withoutsacrificing rate requirement. We observe similar results of fMM[(1− ε)R] in Fig. 2.7(a).

Comparing the average delay of our solutions with the optimal. Fig. 2.4(b) givesthe average delay gap of fSO[(1− ε)R] as compared to that of fSO(R), where empirically theaverage delay of fSO[(1 − ε)R] is non-decreasing with the flow rate ratio (1 − ε), implyingthat the minimal average delay is non-decreasing with the flow rate requirement. Fig. 2.5(b)presents the average delay gap of fSF[(1 − ε)R] as compared to that of fSO(R), where it isobvious that the average delay of fSF[(1− ε)R] is always upper bounded by that of fSO(R).Interestingly, different from the average delay of fSO[(1−ε)R] given in Fig. 2.4(b), the averagedelay of fSF[(1 − ε)R] is not monotonic with the flow rate ratio (1 − ε). From Fig. 2.6(b)

24

Table 2.3: Suppose the sender is IS, the receiver is IL, and the rate requirement is 10. Wegive corresponding delay results of the conceivable baseline with different θ.

θ 0.01 0.05 0.1 0.2

Max-delay 0.63 0.65 0.68 1.17Avg-delay 0.63 0.64 0.67 0.77

(resp. Fig. 2.7(b)), we observe that the empirical average delay gap of fNE[(1 − ε)R] (resp.of fMM[(1− ε)R]) is substantially smaller than the theoretical average delay gap (1− ε, 1/ε).

For a small traffic loss of ε = 0.03, the average delay gap is 0.972 for both fNE[(1 − ε)R]and fMM[(1 − ε)R]. This again shows the benefit of minimizing the two delay metrics bysacrificing flow rate: the empirical average delay gap is 1.037 for both fNE(R) and fMM(R)when we do not sacrifice traffic.

Overall, empirically we verify that better performances on the two delay metrics can beachieved when we can sacrifice a small portion (e.g., ε = 0.03) of flow rate requirement.Note that 3% traffic loss is very acceptable for video conferencing [92].

2.6.2 Comparing Our Solutions with a Conceivable Baseline

In the previous section we empirically verify proposed delay gaps of our solutions that support(1−ε)-fraction of the rate requirement, as compared to the optimal delay performance underfull rate requirement. In this section, we further compare our solutions under reduced raterequirement with a conceivable delay-aware baseline under full rate requirement.

A conceivable baseline fB(R). To send R flow rate from a sender to a receiver with lowdelay, a conceivable approach is to iteratively assigns a small fraction of the flow rate tothe fastest path till all the flow rate are routed. We define fB(R) as the solution to thisconceivable baseline. To be specific, in each iteration this conceivable approach (i) obtainsthe fastest sender-to-receiver path, then (ii) assigns a rate of (θ ·R) (θ ∈ (0, 1)) to the path,and (iii) updates the delay of all links e ∈ E.

It is clear that the conceivable baseline can obtain better delay performance with smaller θ(Tab. 2.3 gives an illustrative example). We set θ = 0.01 in the following simulations.

To be consistent, we also give simulation results of fB[(1− ε)R], which is the solution to theconceivable baseline subject to a smaller flow rate requirement of (1− ε) ·R. We do not givethe results of fMM[(1 − ε)R] in this section, due to the following two concerns: (i) figuringout fMM[(1− ε)R] is time-consuming since MM is NP-hard, and (ii) θ = 1% is small enoughfor fB[(1− ε)R] to be a good approximation to fMM[(1− ε)R].

Suppose the sender is IS and the receiver is IL. First, for each ε ∈ (0, 0.1) with a step of 0.01where R is fixed to be 10, Fig. 2.8 gives empirical results which compare the delay of solutions

25


Figure 2.8: Ratios comparing the delays of fB[(1 − ε)R], fSO[(1 − ε)R], fSF[(1 − ε)R], andfNE[(1− ε)R], respectively with that of fB(R). We set s = IS, t = IL, R = 10, and vary ε.


Figure 2.9: Ratios comparing the delays of fB[(1 − ε)R], fSO[(1 − ε)R], fSF[(1 − ε)R], andfNE[(1− ε)R], respectively with that of fB(R). We set s = IS, t = IL, ε = 0.03, and vary R.

with a sacrificed rate requirement of (1 − ε) · R with that of the baseline fB(R) subject tothe full rate requirement of R. From the figure we observe that (i) solutions with sacrificedrate requirement obtain a smaller maximum delay (resp. smaller average delay) than fB(R),as the maximum delay ratio (resp. average delay ratio) is below 1 for many instances; (ii)a smaller maximum delay gap (resp. smaller average delay gap) can be achieved as thesacrificing ratio (i.e., ε) becomes larger; (iii) the maximum delay (resp. average delay) offSO[(1− ε)R] and fSF[(1− ε)R] is larger than (resp. smaller than) that of fNE[(1− ε)R] andfB[(1 − ε)R]; and (iv) the maximum delay (resp. average delay) of fNE[(1 − ε)R] is slightlysmaller than that of fB[(1− ε)R].

Next, for each rate requirement R ∈ (4, 14) with a step of 1 where ε is fixed to be 0.03,Fig. 2.9 shows the empirical results which compare the delay of solutions supporting a flowrate of (1 − ε) · R with that of the baseline fB(R). We observe that (i) smaller maximum

26


Figure 2.10: Ratios comparing the delays of fB[(1 − ε)R], fSO[(1 − ε)R], fSF[(1 − ε)R], andfNE[(1− ε)R], respectively with that of fB(R). We set s = PT, t = EE, R = 15, and vary ε.


Figure 2.11: Ratios comparing the delays of fB[(1 − ε)R], fSO[(1 − ε)R], fSF[(1 − ε)R], andfNE[(1− ε)R], respectively with that of fB(R). We set s = PT, t = EE, ε = 0.03, and vary R.

delay gap and smaller average delay gap can be achieved as the flow rate requirement (i.e.,R) becomes larger; (ii) comparing solutions fSO[(1 − ε)R] and fSF[(1 − ε)R] with solutionsfNE[(1 − ε)R] and fB[(1 − ε)R], the maximum delay of the former is larger than that ofthe latter while the average delay of the former is smaller than that of the latter; and (iii)fNE[(1− ε)R] performs slightly better than fB[(1− ε)R], achieving both a smaller maximumdelay and a smaller average delay.

We give simulation results of another sender-receiver pair (we assume that sender is thebottom-left PT, and receiver is the top-right EE) in Fig. 2.10 and 2.11. We observe thatFig. 2.10 is similar to Fig. 2.8, and Fig. 2.11 is similar to Fig. 2.9.

Overall, we observe that if one can accept a small loss of rate requirement, he/she can obtain amuch smaller maximum delay and a much smaller average as compared to the respective delayperformance without traffic loss. For example, for a 3% traffic loss when the rate requirement

27

is large, Fig. 2.9 and 2.11 show that over 10% reduction can be achieved simultaneously on themaximum delay and on the average delay for each of the four solutions subject to sacrificedrate requirement, i.e., for fB[(1 − ε)R], fSO[(1 − ε)R], fSF[(1 − ε)R], and fNE[(1 − ε)R],as compared to fB(R) that is subject to full rate requirement. Furthermore, we observethat fSO[(1− ε)R] and fSF[(1− ε)R] empirically outperform fB[(1− ε)R] and fNE[(1− ε)R]in minimizing average delay, but fB[(1 − ε)R] and fNE[(1 − ε)R] empirically outperformfSO[(1−ε)R] and fSF[(1−ε)R] in minimizing maximum delay. Besides, fNE[(1−ε)R] is slightlybetter than fB[(1−ε)R] both in minimizing average delay and in minimizing maximum delay.The average delay of fSO[(1− ε)R] is smaller than that of fSF[(1− ε)R], while the maximumdelay of fSO[(1− ε)R] is larger than that of fSF[(1− ε)R].

2.7 Extension to Multiple-Unicast Networking

In previous sections we design three single-unicast solutions, i.e., fSF[(1− ε)R], fNE[(1− ε)R],and fMM[(1 − ε)R], all of which can minimize the two delay metrics both within bi-criteriaconstant-ratio gaps to the optimal. In this section, we generalize our solutions to the multiple-unicast networking scenario.

2.7.1 System Model for Multiple-Unicast Networking

Suppose we are given K sender-receiver pairs (sk, tk) : k = 1, 2, ..., K. For each k ∈1, 2, ..., K, we require sk ∈ V to use multiple paths to stream a flow at a fixed rate of Rk

to tk ∈ V \sk. We define Pk as the set of simple paths from sk to tk. For easier reference,we further define

Rmin , min1≤k≤K

Rk, R ,K∑k=1

Rk, P , ∪Kk=1 Pk.

A multiple-unicast flow solution f = fk : k = 1, 2, ..., K is defined as the assigned ratesover P , where each fk is a single-unicast flow defined as the assigned rates over Pk. Wedefine xek as the aggregate assigned rate of unicast k over edge e,

xek ,∑

p∈Pk:e∈p

xp, ∀k = 1, 2, ..., K.

We define xe ,∑K

k=1 xek. The delay of a path p ∈ Pk under a single-unicast flow fk is

dp(fk)

,∑e∈p

De(xe).

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The total rate of a single-unicast flow fk is∣∣fk∣∣ ,∑p∈Pk

xp, ∀k = 1, 2, ..., K.

Similar to Equations (2.7), (2.8), and (2.9), we define the maximum delay, total delay, andaverage delay, respectively for each single-unicast flow fk as follows

M(fk), max

p∈Pk:xp>0dp(fk),∀k = 1, 2, ..., K,

T(fk),∑p∈Pk

dp(fk)· xp =

∑e∈E

De(xe) · xek,∀k = 1, 2, ..., K,

A(fk), T

(fk)/∣∣fk∣∣ ,∀k = 1, 2, ..., K.

We consider multiple-unicast problems MM, SO, and NE that are defined by existing stud-ies [14, 15, 73]. First, we define the maximum delay, total delay, and average delay of amultiple-unicast flow f = fk : k = 1, ..., K similarly as those proposed by [14,15,73]

M(f) , max1≤k≤K

M(fk), T (f) ,

K∑k=1

T(fk),

A(f) ,K∑k=1

T(fk)/

K∑k=1

∣∣fk∣∣ .Based on the three delay metrics, studies [14,15,73] then define MM, SO, and NE under themultiple-unicast setting in the following

MM : minfM(f) s.t.

∣∣fk∣∣ = Rk,∀k = 1, 2, ..., K,

SO : minfA(f) s.t.

∣∣fk∣∣ = Rk, ∀k = 1, 2, ..., K,

NE : find a Nash flow f s.t.∣∣fk∣∣ = Rk,∀k = 1, 2, ..., K.

2.7.2 Existing Delay-Gap Results in the Multiple-Unicast Setting

We have summarized existing delay gaps of MM, SO, and NE under the single-unicast settingin Tab. 2.1. Now we look at their delay performances under the multiple-unicast setting andsummarize corresponding results in Tab. 2.4.

A critical observation on fNE(R) in the multiple-unicast setting is as follows [15, Thm. 5.2]:

Rk · A(fkNE(R)

)≤ R · A (fNE(R)) , ∀k = 1, 2, ..., K. (2.28)

Following a similar proof, it is easy to generalize the inequalities in (2.28) from the Nashflow to an arbitrary multiple-unicast flow f = fk : k = 1, 2, ..., K.

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Table 2.4: Existing delay gaps of SO, NE, and MM, under the multiple-unicast scenario [14,15].

Average delay gap Maximum delay gapfSO(R) 1 γ(L) · (R/Rmin)fNE(R) σ(L) σ(L) · (R/Rmin)fMM(R) σ(L) · (R/Rmin) 1

Lemma 2. Under the multiple-unicast scenario, for any flow f = fk : k = 1, 2, ..., Kwhere |fk| = Rk,∀k = 1, 2, ..., K, and |f | = R, we have the following

A(fk)≤ (R/Rmin) · A(f), ∀k = 1, 2, ..., K.

Proof. It follows a similar proof to [15, Thm. 5.2]. Specifically, due to T (fk) ≤ T (f),considering the definition of the total delay, the following holds

A(fk)≤ (R/Rk) · A(f), ∀k = 1, 2, ..., K,

which implies that

A(fk)≤ (R/Rmin) · A(f), ∀k = 1, 2, ..., K.

With the inequalities in (2.28), Correa et al. [14,15] prove that for NE, its average delay gap isσ(L) and its maximum delay gap is σ(L)·(R/Rmin) in the multiple-unicast setting. AlthoughCorrea et al. [14,15] do not extend the maximum delay gap of SO (resp. average delay gap ofMM) which is γ(L) (resp. σ(L)) in a single-unicast setting to the multiple-unicast scenario,with Lem. 2 we can establish the following two theorems.

Theorem 9. Under the multiple-unicast scenario, the following holds comparing the averagedelay of fMM(R) with the optimal

A(fMM(R)) ≤ σ(L) · (R/Rmin) · A∗(R).

Proof. It holds due to the following

A(fMM(R)) ≤M(fMM(R)) ≤M(fNE(R))(a)= A

(f iNE(R)

)(b)

≤ (R/Rmin) · A(fNE(R)) ≤ σ(L) · (R/Rmin) · A∗(R),

where in (a) we assume the maximum delay of fNE(R) is achieved by the unicast i, and (b)holds due to Lem. 2.

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Theorem 10. Under the multiple-unicast scenario, the following holds comparing the max-imum delay of fSO(R) with the optimal

M(fSO(R)) ≤ γ(L) · (R/Rmin) · M∗(R).

Proof. It holds due to the following

M(fSO(R))(a)

≤ γ(L) · A(f iSO(R))(b)

≤ γ(L) ·RRmin

· A(fSO(R))

≤ γ(L) ·RRmin

· A(fMM(R)) ≤ γ(L) ·RRmin

· M(fMM(R)),

where (a) comes from [15, Thm. 4.2], assuming that the maximum delay of fSO(R) is achievedby the unicast i, and (b) holds due to Lem. 2.

In addition to existing studies [14,15], Roughgarden [73] poses a question of whether (|V |−1)remains to be a maximum delay gap of NE under the multiple-unicast setting, yet withoutanswering it.

Overall under the multiple-unicast setting, solving MM, NE, and SO all can minimize themaximum delay and the average delay with approximation ratios that depend on link delayfunctions and flow rate requirements. As discussed in our Sec. 2.2, those problem-dependentgaps can be arbitrarily large for certain delay functions. Therefore, none of the optimalsolutions to the three problems can minimize the two delay metrics within bounded-ratiogaps to the optimal.

2.7.3 Extending Our Results to the Multiple-Unicast Scenario

Similar to the definitions introduced in Sec. 2.3.3, now we generalize our solutions fMM[(1−ε)R] and fNE[(1− ε)R] to the multiple-unicast setting

• fMM[(1 − ε)R]: an optimal solution to MM, where the unicast k is subject to a flowrate requirement of (1− ε) ·Rk, for each k = 1, 2, ..., K.

• fNE[(1− ε)R]: an optimal solution to NE, where the unicast k is subject to a flow raterequirement of (1− ε) ·Rk, for each k = 1, 2, ..., K.

Based on their definitions, it is clear that∣∣fkMM[(1− ε)R]∣∣ = (1− ε) ·Rk, ∀k = 1, 2, ..., K,∣∣fkNE[(1− ε)R]∣∣ = (1− ε) ·Rk, ∀k = 1, 2, ..., K.

For fSF[(1− ε)R], we have two different definitions under the multiple-unicast scenario

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• fSF-1[(1−ε)R]: same to Algorithm 1, it is a solution returned after we iteratively deletea total of (1− ε) ·R rate from the slowest flow-carry paths of fSO(R).

• fSF-2[(1 − ε)R]: similar to Algorithm 1, it is a solution returned after we iterativelydelete a total of (1− ε) ·Rk rate from the slowest flow-carrying paths of the unicast kof fSO(R), for each k = 1, 2, ..., K.

It is clear that we have the following

|fSF-1[(1− ε)R]| = (1− ε) ·R,

i.e., fSF-1[(1 − ε)R] can support (1 − ε)-fraction of the aggregate rate requirement of allunicasts. However, we remark that there may exist an unicast k ∈ 1, 2, ..., K where|fkSF-1[(1− ε)R]| < (1− ε) ·Rk. In comparison, according to the definition of fSF-2[(1− ε)R],we have ∣∣fkSF-2[(1− ε)R]

∣∣ = (1− ε) ·Rk, ∀k = 1, 2, ..., K.

As we note that [74, Thm. 3.2] holds in the multiple-unicast setting, we have the following.

Theorem 11. Under the multiple-unicast scenario, the following holds for fNE[(1− ε)R]

A(fNE[(1− ε)R]) ≤ (1/ε) · A∗(R),

M(fNE[(1− ε)R]) ≤ (1/ε) · (R/Rmin) · M∗(R).

Proof. The average delay gap follows a similar proof to Thm. 5, and is an adaptation of [74,Thm. 3.2]. The maximum delay gap follows a similar proof to Thm. 6, based on Lem. 2.

For fMM[(1− ε)R], we have the following.

Theorem 12. Under the multiple-unicast scenario, the following holds for fMM[(1− ε)R]

A(fMM[(1− ε)R]) ≤ (1/ε) · (R/Rmin) · A∗(R),

M(fMM[(1− ε)R]) ≤ M∗(R).

Proof. The average delay gap follows a similar proof to Thm. 7, based on Lem. 2. Themaximum delay gap holds straightforwardly due to the non-decreasing property of link delayfunctions.

For fSF-1[(1− ε)R], we have the following.

Theorem 13. Under the multiple-unicast scenario, the following holds for fSF-1[(1− ε)R]

A(fSF-1[(1− ε)R]) ≤ A∗(R),

M(fSF-1[(1− ε)R]) ≤ (1/ε) · M∗(R).

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Table 2.5: Bi-criteria delay gaps of our proposed solutions in the multiple-unicast setting.Average delay gap Maximum delay gap

fNE[(1− ε)R] (1− ε, 1/ε) (1− ε, R/(ε ·Rmin))fMM[(1− ε)R] (1− ε, R/(ε ·Rmin)) (1− ε, 1)fSF-1[(1− ε)R]∗ (1− ε, 1)∗ (1− ε, 1/ε)∗fSF-2[(1− ε)R] (1− ε, R/Rmin) (1− ε, R/(ε ·Rmin))

Note. ∗: fSF-1[(1− ε)R] can only support (1− ε)-fraction of the aggregate rate requirement R, while all theother three solutions can support (1− ε)-fraction of the rate requirement Rk for each unicast k = 1, ...,K.

Proof. The average delay gap follows a similar proof to Thm. 3. The maximum delay gapfollows a similar proof to Thm. 4.

For fSF-2[(1− ε)R], we have the following theorem.

Theorem 14. Under the multiple-unicast scenario, the following holds for fSF-2[(1− ε)R]

A(fSF-2[(1− ε)R]) ≤ (R/Rmin) · A∗(R),

M(fSF-2[(1− ε)R]) ≤ (1/ε) · (R/Rmin) · M∗(R).


Our results are summarized in Tab. 2.5. We observe that fSF-1[(1 − ε)R] can minimize thetwo delay metrics both within bi-criteria constant-ratio gaps to the optimal, while all theother three solutions can minimize the two delay metrics both within bi-criteria problem-dependent-ratio gaps to the optimal. Here the problem-dependent ratio only depends onflow rate requirements of individual unicasts, and is independent to link delay functions.For general network topologies and arbitrary link delay functions, comparing Tab. 2.4 withTab. 2.5, existing delay gaps depend on link delay functions, and hence can be unbounded;in comparison, our bi-criteria delay gaps are independent to link delay functions, and henceare always bounded. Overall, our solutions all can minimize the two delay metrics bothwithin bi-criteria bounded-ratio gaps to the optimal in the multiple-unicast setting.

2.8 Chapter Summary

In this chapter we consider the scenario where in a multi-hop network a sender requiresto use multiple paths to stream a flow at a fixed rate to a receiver. We consider a delaymodel where the experienced delay of passing a link is assumed to be a non-decreasing,non-negative, and differentiable function of the link aggregate transmission rate. Our ob-jective is to minimize two E2E delay metrics, i.e., the experienced average sender-to-receiverdelay and the experienced maximum sender-to-receiver delay, from the perspective of three

33

fundamental network delay optimization problems. The three problems are (i) MM thatminimizes maximum delay, (ii) SO that minimizes average delay, and (iii) NE that finds theNash equilibrium.

For general network topologies and arbitrary link delay functions, existing results suggestthat a flow cannot simultaneously minimize the two delay metrics both within bounded-ratio gaps to the optimal. In comparison, we construct three solutions, each of which canminimize the two delay metrics both within a (1/ε)-ratio gap to the optimal, at a cost ofonly streaming (1− ε)-portion of the flow, for any ε ∈ (0, 1). We prove that the ratio (1/ε)is at least near-tight, and is independent to the link delay function and network topology.Moreover, we generalize our solutions to the multiple-unicast setting, where we prove thateach of our solutions minimizes the two delay metrics both within a bounded-ratio gap of(R/(Rmin · ε)) to the optimal. Here R (resp. Rmin) is the aggregate (resp. minimum) flowrate requirement of all individual unicasts.

We empirically evaluate our theoretical results using extensive simulations by supportingvideo-conferencing traffic. The empirical delay gaps are much smaller than the theoreticalcounterparts, and our solutions can achieve over 10% reduction both on the maximum delayand on the average delay, at a cost of only losing 3% traffic, as compared to a conceivabledelay-aware baseline without traffic loss.

Chapter 3

Minimize Age-of-Information withThroughput Requirements

3.1 Introduction

AoI is an important delay-aware metric for services that require timely and periodicallytransmissions. In the networking area, it is first proposed in [43] for a vehicular networkstudy to our best knowledge. To be specific, Kaul et al. [43] define AoI as follows: if a newpacket is received by a receiver, the AoI of the receiver will be the elapsed time since thepacket generation; otherwise, the AoI of the receiver grows linearly. By its definition, AoIcaptures the information “freshness” in terms of the receiver. In this chapter in a single-unicast communication setting, we consider a periodic transmission task where a senderrequires to use multiple paths to send a batch of data (packets) to a receiver periodically.Our objective is to minimize either the peak AoI or the average AoI, subject to both aminimum and a maximum throughput requirement. Our design spaces include multi-pathrouting optimization as well as throughput optimization. We assume the size of the databatch to be fixed, and thus the throughput only varies with the task activation period.

Our study is motivated by the increasingly need of supporting real-time video cognition tasksin mobile devices, with the help of an edge computing platform. We note that current mobiledevices can process images. For example, DeepEye [62] is a prototype wearable camera tosupport life-logging and vision assistance. Deep learning is often involved to make smartdecisions.

However, it is well-known that mobile device is resource-constrained, but deep learning isresource-heavy. Hence it is almost impossible for a mobile device to process images in areal-time manner. For example on a typical Android phone, the video can only be processedat a rate less than 1 FPS using Tensorflow’s deep learning model [70]. The widely-adoptedsolution is to process some images locally on phones, while offload other images to nearby

34

35

Table 3.1: Comparing our AoI work with existing AoI studies.[43,44,82] [81] [37] [85] [42] [83,84] [86] Our work

Objective ofOptimization

Minimize peak AoI 7 3 3 3 7 3 3 3

Minimize average AoI 3 3 7 3 3 3 3 3

Design Space ofOptimization

Multi-path routing strategy 7 7 7 7 7 7 7 3

Information generation rate 3 7 3 3 3 7 3 3

Link scheduling policy 7 7 7 7 3 3 3 7

Queuing disciplines 7 3 3 3 7 7 7 7

Other Results Compare AoI with delay 3 7 3 3 7 7 7 3

powerful edge servers for processing. For example, when running the video recognitionapplication designed by Ran et al. [70] locally on the phone for 30 minutes, consumes 25%battery and processes images at a 5 FPS rate; while when running remotely on an edgeserver, it consumes 15% battery and processes images at a 9 FPS rate.

We note from [70] that networking delay dominates the delay of offloading images to edgeserves for processing. Therefore, in order to timely and efficiently utilize networking re-sources, a delay-aware offloading algorithm is highly critical. With our proposed algorithms,images generated by mobile devices can be timely offloaded to edge servers through anarbitrary network topology.

We compare existing AoI studies with ours in Tab. 3.1. We differ from existing AoI studiesboth in the problem design space and in the AoI definition: we are the first to consider themulti-path routing optimization as a design space of minimizing AoI, to our best knowledge;besides, existing studies assume a packet-based definition of AoI, where AoI can be updatedby receiving any packet. The packet-based AoI is reasonable in status update systems. Incomparison, we assume a task-based definition of AoI, as it is reasonable that only afterreceiving a complete data batch, the receiver can reconstruct information of one task periodand hence update AoI to be the elapsed time since the batch generation. An example of ourbatch-based AoI is given in Fig. 3.1, assuming slotted data transmissions.

We further remark that our problem differs from existing delay-aware multi-path commu-nication studies, in that the task activation period in our AoI minimization problem is adecision variable, while that in existing maximum delay minimization problems is fixed. Wepresent the detailed comparison between our problem and existing delay-aware multi-pathcommunication problems later in Sec. 3.2.

Overall in this chapter, we study fundamental networking problems of minimizing peak/averageAoI subject to both a minimum and a maximum throughput requirement, for periodicallysending a batch of data in a single-unicast setting. Our design space includes the multi-path routing optimization as well as the throughput optimization. We claim the followingcontributions.

Comparing the problem of minimizing maximum delay with that of minimizing peak/averageAoI, (i) we prove that the throughput achieved by the optimal solution to the former can differ

36

s r

AoI

Time Slot0 3 6-3-6

1

2

3

Link delay is 1 time slot

Solution: s streams 1 packet to (s, r), at each time

slot k×3 and k×3+1 (k is an arbitrary integer)

Problem: a task requires to send 2 packets from s to

r, at each time slot k×3 (k is an arbitrary integer)

Figure 3.1: An illustrative example of the batch-based AoI. Suppose that sender s generatesa batch of two packets at each slot 3k,∀k ∈ Z. It sends the two packets one-by-one overlink (s, r) to the receiver r; the link transmission incurs one-slot delay. As r receives all thetwo packets in a batch at each slot 3k + 2,∀k ∈ Z, the batch-based AoI becomes 2 at eachslot 3k + 2, i.e., the elapsed time since the batch generation. The batch-based AoI increaseslinearly at other slot.

from, but is always bounded above by, that of the optimal solution to the latter (Lem. 5),verifying our observation that AoI jointly considers the achieved throughput and experiencedmaximum delay; (ii) we show that the optimal solution to the former can be suboptimal tothe latter (Lem. 5), but the optimality gap is bounded (Lem. 6).

Comparing the problem of minimizing average AoI with that of minimizing peak AoI, (i) weprove that the average-AoI-optimal solution can have suboptimal peak AoI, and vice versa,but both optimality gaps are bounded (Lem. 7). (ii) We show that the achieved throughput(resp. experienced maximum delay) of the optimal solution to the former can differ from,but is always bounded above by, that of the optimal solution to the latter (Lem. 7). We thusobserve that the peak AoI minimization problem may carry more flavor on throughput andless on maximum delay than its average AoI minimization counterpart.

We argue that it is highly non-trivial to minimize peak/average AoI, because (i) we showthat both minimal average AoI and minimal peak AoI are non-convex, non-concave, andnon-monotonic with achieved throughput (Lem. 11), and (ii) both problems are proven tobe weakly NP-hard (Lem. 8), as we design algorithms for minimizing peak/average AoIoptimally in pseudo-polynomial time complexities (Sec. 3.5.3).

We further leverage our understanding on the comparison between the maximum delayand AoI to develop an approximation framework (Thm. 16). Given any polynomial-timeα-approximation algorithm of minimizing maximum delay, our framework can adopt it toconstruct an (α + c)-approximate solution of minimizing peak/average AoI in a polynomial

37

time, where c is a constant dependent on throughput requirements.

We carry out extensive simulations with different network topologies to evaluate our ap-proaches (Sec. 3.7). Empirically if the range of task activation period increases by 1, (i) ouroptimal algorithm reduces AoI by 3% than our approximation framework; however, (ii) ourapproximation framework reduces running time by 100% than our optimal algorithm.


We have compared our AoI study with existing AoI studies in Tab. 3.1. In this sectionwe further compare our AoI study which is delay-sensitive with existing maximum delayminimization studies.

Our problem is subject to both a maximum and a minimum throughput requirement. Thusthe task activation period (i.e., the size of data batch over achieved throughput) is a decisionvariable. In contrast, existing maximum delay minimization problems all assume a fixedtask activation period to our best knowledge. Those existing problems include min-max-delay flow problem and the quickest flow problem.

Min-max-delay flow problem [54, 64]. Given an amount of data, it finds multiple sender-to-receiver paths such that the aggregate path bandwidth is no smaller than the given amountof data, and the maximum path delay is minimized. It assumes one unit of time for the taskactivation period. Misra et al. [64] prove that the min-max-delay flow problem is NP-hardunder our system model.

Quickest flow problem [53, 76]. It minimizes the time for a sender to use multiple paths tosend a given amount of data to a receiver. It assumes an infinitely large task activationperiod, and is polynomial-time solvable under our system model [53].

Suppose D is the given amount of data and T is the task activation period. Note thatunder our system model, the min-max-delay flow problem assumes T = 1, the quickest flowproblem assumes T = +∞, but our problem assumes D/Ru ≤ T ≤ D/Rl where Rl (resp.Ru) is our minimum (resp. maximum) throughput requirement. Thus from the perspectiveof the throughput R, the min-max-delay flow problem assumes R = D, the quickest flowproblem assumes R → 0, but our problem assumes 0 < Rl ≤ R ≤ Ru ≤ D. Accordingto our Lem. 4 introduced later, given Rl = Ru, minimizing AoI is equivalent to minimizingmaximum delay. Therefore, our problem covers the min-max-delay flow problem and thequickest flow problem as special cases. We further remark that although existing studieshave designed approximation algorithms, e.g., [64], for the min-max-delay flow problem anddeveloped exact algorithms, e.g. [53], for the quickest flow problem, it is unclear of solvingour problems even given 0 < Rl = Ru ≤ D. Moreover, according to our Lem. 5 introducedlater, given 0 < Rl < Ru ≤ D, minimizing AoI differs from minimizing maximum delay.Overall, we observe that it is highly non-trivial to solve our AoI-minimization problems.

38

Table 3.2: Summary of important notations used in our AoI study.D The amount of data in the batch

T (f) Task activation period of a solution f

Λp(f) (resp.Λa(f), M(f))

Peak AoI (resp. Average AoI, Maximum delay) of f

Rl (resp. Ru) The minimum (resp. maximum) throughput requirementΛRp (resp. ΛR

a ,MR)

Minimal peak AoI (resp. Minimal average AoI, Minimal maximum delay)that supports a throughput of R

Rp (resp. Ra,Rm)

The optimal throughput of our problem of minimizing peak AoI(resp. minimizing average AoI, minimizing maximum delay)

Many other delay-aware communication studies also exist in the literature. Hou et al. [35]design scheduling policies that can figure out whether multiple QoS constraints are feasiblefor a given set of senders. In [36], Hou et al. extend their previous work and study the utilitymaximization problem. By assuming a more general traffic pattern, Deng et al. [20] studya similar timely wireless flow problem. Those studies [20, 35, 36] are less relevant with ourwork, because they focus on optimizing the wireless link scheduling policy, while our focusis to optimize the multi-path routing strategy as well as the achieved throughput.

3.3 Preliminary

3.3.1 System Model

We use a directed graph G , (V,E) to model a multi-hop network. We denote the numberof nodes by |V | and the number of links by |E|. We consider a slotted data transmissionsmodel in this chapter. Each link e can send an amount of data bounded above by the linkbandwidth be ∈ R, be ≥ 0 to it, incurring a delay of de ∈ Z+ slots. In addition, we assumethat an arbitrary amount of data can be held by each node v ∈ V at each slot. For easierreference, we use “at time t” or “at slot t” to refer to “at the beginning of the time slot t”.We consider a periodic transmission task in this chapter. Specifically, given that the taskactivation period is T ∈ Z+, there will be D ∈ R, D > 0 amount of data generated at asender s ∈ V at slot kT for each k ∈ Z, and the task requires the sender to use multiple pathsto send them to a receiver r ∈ V \s. We assume no data loss during transmission, andhence it is clear that in our setting for a task activation period of T , the incurred throughputis D/T .

We focus on “fresh” multi-path routing solutions that are periodically repeated. We notethat the metric AoI can evaluate the information “freshness”. Here AoI depends on theE2E delay (see our formula (3.4) presented later). The networking delay is known to becomposed of transmission delay, propagation delay, and queuing delay. Note that the slotted

39

data transmission model can consider all these delays (refer to Appendix A.5), similar to thediscussions in [6]. Suppose P is the set of all sender-to-receiver simple paths, and |p| is thenumber of nodes on the path p ∈ P . A periodically repeated solution f can be defined indifferent ways, e.g., we can define f as the data allocated over P at the slot offset ~U ,

f ,xp(~u) ≥ 0 : ∀p ∈ P, ∀~u ∈ ~U

, (3.1)

where we have the following definition for ~U : let us assume p = 〈v1, v2, ..., v|p|〉 ∈ P to be anarbitrary path, where vi ∈ V, 1 ≤ i ≤ |p| are nodes belonging to p and ei−1 = (vi−1, vi) ∈E, 2 ≤ i ≤ |p| are the links on p, with v1 = s and v|p| = r. Any offset ~u ∈ ~U correspondingto p is ~u = 〈u0, u1, u2, ..., u|p|〉, where we have the following with u0 = 0 and de0 = 0

ui ∈ Z and ui ∈ [ui−1 + dei−1, ui−1 + dei−1

+ U ], ∀i = 1, 2, ..., |p|. (3.2)

We assume that T (f) is the task activation period of f . Now each positive xp(~u) of f suggeststhat we should send xp(~u) amount of data to (vi, vi+1) at the offset ui, i.e., send xp(~u) amountof data of the period starting at the slot k · T (f) to the link (vi, vi+1) at the slot k · T (f)+ui.Note that in the definition (3.2), the inequalities ui − dei−1

− ui−1 ≤ U,∀i = 1, 2, ..., |p|restricts the node data-holding delay to be bounded above by U slots. In general, we haveU = +∞ for our problems. Later in Lem. 3, we further prove that if we are interested insolutions with a task activation period of T , setting U = T − 1 is large enough to solve anyproblem instance that is theoretically feasible.

Based on each positive xp(~u) of a solution f , according to the aforementioned definitions, it iseasy for us to obtain (i) the beginning offset of sending those data to ei ∈ p, i = 1, ..., |p| − 1,denoted as Bp(~u, ei),

Bp(~u, ei) = ui,

and (ii) the experienced sender-to-receiver delay of those data, denoted as Ap(~u),

Ap(~u) = Bp(~u, e|p|−1) + de|p|−1= u|p|−1 + de|p|−1

.

which is the summation of the offset u|p|−1 when the data xp(~u) is pushed onto the last link(v|p|−1, v|p|) that belongs to the path p, and the link delay of the last link.

Maximum delay, denoted by M(f), is known to be one critical networking performancemetric of f . It is the time of sending a complete data batch from the sender to the receiver,i.e.,

M(f) , max∀p∈P,∀~u∈~U : xp(~u)>0

Ap(~u). (3.3)

We define I(f, t) as the AoI of f at time t. It is a maximum-delay-dependent networkingperformance metric, which is the elapsed time since the last received batch generation, i.e.,

I(f, t) , t− πt(f), (3.4)

40

where πt(f) is the generation time of most recently received data batch in terms of thereceiver by the slot t, or equivalently the starting time of the most recent task period withall its R amount of data arriving at the receiver r no later than the time t, i.e.,

πt(f) , maxk∈Zk · T (f) : k · T (f) +M(f) ≤ t .

3.3.2 Problem Definition

Our objective in this chapter is to minimize either (i) the average value of AoI, or (ii) thepeak value of AoI. We denote Λa(f) as the average AoI of f , which is

Λa(f) ,∑t∈Z

I(f, t) /∑t∈Z

1. (3.5)

Similarly, we denote Λp(f) as the peak AoI of f , which is

Λp(f) , maxt∈ZI(f, t). (3.6)

Our AoI minimization problems are subject to link bandwidth constraints, a minimumthroughput requirement, and a maximum throughput requirement. The minimum (resp.maximum) throughput requirement restricts that the achieved throughput of a solution belower bounded by an input Rl ∈ R (resp. upper bounded by an input Ru ∈ R), i.e.,∑

∀p∈P

∑∀~u∈~U

xp(~u) = D,Rl ≤ D/T (f) ≤ Ru, and T (f) ∈ Z+. (3.7)

Due to that T (f) ∈ Z+ holds in our slotted data transmission model, it is fair to assumeD/Rl ∈ Z+ and D/Ru ∈ Z+.

Given a solution f , we denote xe(i) as the aggregate amount of data sent to the link e ∈ Eat the time offset i ∈ 0, 1, ..., T (f)−1. Note that 0 ≤ i ≤ T (f)−1, i ∈ Z, considering thatthe periodically repeated property of f implies xe(i + T (f)) = xe(i). The link bandwidthconstraints require xe(i) to be upper bounded by be, i.e., xe(i) ≤ be, for any offset i =0, 1, ..., T (f)− 1 and any link e ∈ E, i.e., the aggregate data sent to each e ∈ E at each slotshall be no greater than be.

It is clear that xp(~u) contributes to xe(i) if and only if there exists k ∈ Z with k · T (f) +Bp(~u, e) = i as well as e ∈ p. The link bandwidth constraints hence can be formulated asfollows ∑

p∈P :e∈p

∑k∈Z,~u:k·T (f)+Bp(~u,e)=i

xp(~u) ≤ be, ∀i = 0, ..., T (f)− 1,∀e ∈ E. (3.8)

Suppose ΛRp (resp. ΛR

a ) is the minimal peak AoI (resp. minimal average AoI) that canbe achieved by any feasible periodically repeated solution that has a throughput of R. In

41

this chapter we study following two AoI minimization problems, given inputs of throughputrequirements Rl and Ru, a sender s ∈ V , a receiver r ∈ V \s, and a network G(V,E),

1. Obtain an optimal throughput Rp ∈ [Rl, Ru], D/Rp ∈ Z+ that minimizes peak AoI,

Rp , arg minRl≤R≤Ru,D/R∈Z+

ΛRp .

and obtain the corresponding optimal periodically repeated solution. This MinimizingPeak AoI problem is denoted by MPA.

2. Obtain an optimal throughput Ra ∈ [Rl, Ru], D/Ra ∈ Z+ that minimizes average AoI,

Ra , arg minRl≤R≤Ru,D/R∈Z+

ΛRa ,

and obtain the corresponding optimal periodically repeated solution. This MinimizingAverage AoI problem is denoted by MAA.

Similar to MPA and MAA, (i) we denote the minimal maximum delay with a throughput ofR byMR, and (ii) we define the problem of Minimizing Maximum Delay (MMD) as to figureout an optimal Rm ∈ [Rl, Ru], D/Rm ∈ Z+ that minimizes maximum delay, and obtainingassociated optimal periodically repeated solution.

Finally, we give a lemma below, which proves that if there is a feasible solution g with thedata-holding delay greater than T (g)− 1 slots for certain node, another feasible solution fmust exist such that the data-holding delay of f is upper bounded by T (f)− 1 slots for allnodes, and we have: (i) the throughput of f is same to that of g, and (ii) both the averageAoI and the peak AoI of f is bounded above by the respective AoI performance of g. Adirect corollary is that for any feasible instance of MPA (resp. MAA), if we are interested insolutions with a task activation period of T , setting U (defined in (3.1)) to be T − 1 is largeenough to solve the instance.

Lemma 3. Given any MPA (or MAA) instance, suppose g is an arbitrary feasible periodicallyrepeated solution. Then we have that another feasible periodically repeated solution f mustexist, further with the following held: T (f) = T (g), Λp(f) ≤ Λp(g), Λa(f) ≤ Λa(g), andui − dei−1

− ui−1 ≤ T (f) − 1 holds for all i = 1, 2, ..., |p|, for each positive xp(~u) (supposep = 〈v1, ..., v|p|〉 and ~u = 〈u0, ..., u|p|〉) of f .

Proof. Refer to Appendix A.6.

3.4 Comparing AoI with Maximum Delay

We note that both AoI and maximum delay are time-sensitive networking performance met-rics. In this section, we theoretically compare MMD, i.e., the maximum delay minimizationproblem, with MPA and MAA, i.e., with the two AoI minimization problems.

42

Let us consider a simple two-node (nodes s and t) one-link (link (s, t)) network. We assumethat d(s,t) = d, b(s,t) = b ≥ D, Ru = D, and Rl = D/Tu given a Tu ∈ Z+. In this simpleinstance one solution can send D data to (s, r) at the offset 0. It is clear that this solutionis feasible if the task activation period is T ≤ Tu. Following this solution, the data batchof the period starting at the slot kT will arrive at the receiver at the slot kT + d. For twodifferent task activation periods T1 and T2 (T1 < T2 ≤ Tu), the solution with T = T1 isequivalent to that with T = T2 in minimizing the maximum delay, as both feasible solutionsobtain a maximum delay of d. In sharp contrast, the solution with T = T1 is better thanthat with T = T2 in minimizing the peak/average AoI, because the peak AoI (resp. averageAoI) is d + T1 − 1 (resp. d + (T1 − 1)/2) for the former, which is smaller than d + T2 − 1(resp. d+ (T2 − 1)/2) that is the respective AoI achieved by the latter, based on our Lem. 4presented later. In fact in this illustrative example, T1 is better than T2 since both experiencethe same delay of streaming the batch of data, but the former achieves a larger throughputthan the latter.

Based on this example, comparing AoI with maximum delay, we learn that the former shouldbe optimized to provide delay-aware routing solutions for periodic transmission services, asAoI is a metric simultaneously considering throughput and maximum delay. We furtherprove below that the solution which minimizes maximum delay can achieve a suboptimalpeak/average AoI, but with bounded optimality loss.

First for an arbitrary solution, the following lemma relates its AoI to its maximum delay.

Lemma 4. The following holds for an arbitrary periodically repeated solution f ,

Λa(f) = M(f) + (T (f)− 1)/2, Λp(f) = M(f) + T (f)− 1.


Based on the above lemma, it is clear that the peak AoI (resp. average AoI) of a feasiblesolution that has a maximum delay of MR and a throughput of R is ΛR

p (resp. ΛRa ). This

also implies that in order to solve MPA and MAA given Rl = Ru, we can resort to solvingthe corresponding MMD instead. Nevertheless, as discussed in Sec. 3.2, only special cases ofMMD with Rl = Ru, i.e., the min-max-delay flow problem (Rl = Ru = D) and the quickestflow problem (Rl = Ru → 0), are studied, while it is unclear how to solve MMD even given0 < Rl = Ru ≤ D. Moreover, for general settings with Rl < Ru, in the following lemma weshow that the problems of MPA and MAA are different from the problem of MMD.

Lemma 5. Given any MPA (or MAA, MMD) instance, suppose ~Rp (resp. ~Ra, ~Rm) is theoptimal set of throughputs that minimize the peak AoI (resp. average AoI, maximum delay).For this instance we must have

minRp∈~Rp

Rp ≥ maxRm∈~Rm

Rm, minRa∈~Ra

Ra ≥ maxRm∈~Rm

Rm.

43

And there must exist an instance where we have

minRp∈~Rp

Rp > maxRm∈~Rm

Rm, minRa∈~Ra

Ra > maxRm∈~Rm

Rm.


We observe from Lem. 5 that (i) minimizing AoI differs from minimizing maximum delay, asthe AoI achieved by the maximum-delay-optimal solution can be suboptimal; (ii) the solutionminimizing AoI must obtain a throughput no smaller than that achieved by the solutionminimizing maximum delay. In the lemma below, we further derive theoretically near-tightgaps comparing the AoI of the solution minimizing maximum delay with the optimal AoIperformance.

Lemma 6. Given any MPA (or MAA, MMD) instance, suppose ~Rp (resp. ~Ra, ~Rm) is theoptimal set of throughputs that minimize peak AoI (resp. average AoI, maximum delay). Forthis instance we must have

ΛRmp − ΛRp

p ≤D

Rl

− D

Ru

,∀Rm ∈ ~Rm,∀Rp ∈ ~Rp. (3.9)

ΛRma − ΛRa

a ≤D

2Rl

− D

2Ru

,∀Rm ∈ ~Rm,∀Ra ∈ ~Ra (3.10)

Gap (3.9) is near-tight, as for arbitrary D, Rl, and Ru that meet D > 0, D/Rl ∈ Z+, andD/Ru ∈ Z+, an instance exists where we have

ΛRmp − ΛRp

p ≥ D

Rl

− D

Ru

− 1, ∀Rm ∈ ~Rm,∀Rp ∈ ~Rp.

Gap (3.10) is near-tight similarly, where for the instance we have

ΛRma − ΛRa

a ≥ D

2Rl

− D

2Ru

− 1, ∀Rm ∈ ~Rm,∀Ra ∈ ~Ra.


Overall, MPA and MAA are non-trivial as compared to MMD: (i) the solution that minimizesAoI, instead of the one that minimizes maximum delay, provides a low-delay routing solutionfor periodic communication applications; (ii) for the general scenario where throughput canbe optimized (Rl < Ru), the solution that minimizes AoI can differ from that minimizingmaximum delay; (iii) even in the special scenario with fixed throughput (Rl = Ru), whereit can be proven that the solution which minimizes AoI also minimizes maximum delay,and vice versa, existing delay minimization studies assume special fixed throughput (eitherRl = Ru → 0 or Rl = Ru = D). If the throughput is fixed arbitrarily (0 < Rl = Ru ≤ D), itis still unclear how to solve the corresponding maximum delay minimization problem. In thefollowing sections, we design both an optimal algorithm and an approximation frameworkfor our studied problems MPA and MAA.

44

3.5 Minimizing AoI is NP-Hard in the Weak Sense

In this section we show that MPA can differ from MAA, but both are weakly NP-hard, as wedesign an algorithm for solving them optimally in a pseudo-polynomial time.

3.5.1 MPA differ from MAA, but both are NP-Hard

In the following lemma we prove that MAA of minimizing average AoI can differ from MPAof minimizing peak AoI.

Lemma 7. Given any MPA (or MAA) instance, suppose ~Rp (resp. ~Ra) is the optimal set ofthroughputs that minimize peak AoI (resp. average AoI). For this instance,

1. it holds thatminRp∈~Rp

Rp ≥ maxRa∈~Ra

Ra, minRp∈~Rp

MRp ≥ maxRa∈~Ra

MRa ,

2. and we have the following

ΛRpa − ΛRa

a ≤D

2Rl

− D

2Ru

,∀Rp ∈ ~Rp,∀Ra ∈ ~Ra. (3.11)

ΛRap − ΛRp

p ≤⌊D

2Rl

− D

2Ru

⌋,∀Rp ∈ ~Rp,∀Ra ∈ ~Ra. (3.12)

Moreover, an instance must exist where we have

minRp∈~Rp

Rp > maxRa∈~Ra

Ra, minRp∈~Rp

MRp > maxRa∈~Ra

MRa ,


The above lemma suggests that (i) MAA can be different from MPA; (ii) MPA may carry moreflavor on throughput and less on maximum delay than MAA; (iii) the average AoI (resp. peakAoI) achieved by the optimal solution to MPA (resp. to MAA) is within a bounded optimalityloss as compared to the optimal.

We observe that both MAA and MPA are NP-hard, because (i) MAA and MPA both coverMMD as a special case based on Lem. 4; and (ii) the problem MMD assuming Rl = Ru = D(i.e., the min-max-delay flow problem) is proven to be NP-hard [64].

Lemma 8. Both MAA and MPA are NP-hard.

Proof. Same to the Appendix of [64].

45

s a r

s0

s1 s

2s3

s4

s5

a0

a1

a2

a3

a4

a5

r0

r1

r2

r3

r4

r5

Construct Gexp from G

Figure 3.2: An example of constructing the expanded network Gexp. We assume that d(a,r) =1, d(s,a) = 2, MU = 5, and M = 4.

Next we develop an algorithm which solves MPA (resp. MAA) optimally in a pseudo-polynomial time. It enumerates all possible throughputs R ∈ [Rl, Ru], D/R ∈ Z+ to figureout the Rp that minimizes peak AoI (resp. the Ra that minimizes average AoI), as well asthe optimal routing solution.

3.5.2 Obtaining MR in a Pseudo-Polynomial Time

Given a throughput R with D/R ∈ Z+, now we develop an algorithm that uses an expandednetwork and a scheme of binary search to obtain the minimal maximum delay MR and theassociated periodically repeated solution in a pseudo-polynomial time. Such a solution alsoobtains the minimal peak AoI ΛR

p and the minimal average AoI ΛRa based on our Lem. 4.

Construct an expanded network. Following Algorithm 2, we build Gexp(Vexp, Eexp) whichis an expanded network from G(V,E). Specifically, given an integer upper bound of MR,denoted by MU , we first expand every node v ∈ V to nodes vi, i = 0, ...,MU , where nodevi represents the node v at the slot kT + i in terms of the period which starts at theslot kT, ∀k ∈ Z with T = D/R. We then expand every link e = (v, w) ∈ E to links(vi, wi+de), i = 0, ...,MU − de, where link (vi, wi+de) represents that we can send data to(v, w) at the slot kT + i in terms of the period that starts at the slot kT, ∀k ∈ Z withT = D/R. Finally we construct links (vi, vi+1), i = 0, ...,MU − 1 for each v ∈ V , because inour system model an arbitrary amount of data can be held at each node at each time slot.

Use binary search to obtain MR. Given any integer M with M ≤ MU , the problem ofwhether in G a feasible solution f of our AoI minimization problem exist, with M(f) ≤ Mand T (f) = D/R, can be solved by solving the linear program below which formulates a

46

Algorithm 2 Construct Gexp from G

1: input: G = (V,E), MU

2: output: Gexp = (Vexp, Eexp)3: procedure4: Vexp = Eexp = NULL5: for v ∈ V and i = 0, 1, ...,MU do6: Push node vi into Vexp

7: for e = (v, w) ∈ E and i = 0, 1, ...,MU − de do8: Push link (vi, wi+de) into Eexp

9: for v ∈ V and i = 0, 1, ...,MU − 1 do10: Push link (vi, vi+1) into Eexp

11: return Gexp = (Vexp, Eexp)

maximum flow problem in Gexp.

max∑

e′∈Out(s0)

xe′ (3.13a)

s.t.∑

e′∈Out(s0)

xe′ =∑

e′∈In(rM )

xe′ , (3.13b)

∑e′∈Out(v)

xe′ =∑

e′∈In(v)

xe′ ,∀v ∈ Vexp\s0, rM, (3.13c)

∑e′∈e(i)

xe′ ≤ be, ∀e ∈ E,∀i = 0, 1, ..., D/R− 1, (3.13d)

vars. xe′ ≥ 0, ∀e′ ∈ Eexp. (3.13e)

In the above linear program, we denote the set of incoming (resp. outgoing) links ofv ∈ Vexp in Gexp by In(v) (resp. Out(v)). Besides, e(i) is the set of expanded links(vkT+i, wkT+i+de), ∀k ∈ Z in Eexp, where e = (v, w) is a link in E and T = D/R. Wenote that because the aggregate data allocated to e(i) is xe(i)(introduced in (3.8)), the ag-gregate data allocated to e(i) should be upper bounded by the link bandwidth of be. Theobjective in (3.13a) maximizes the streamed data. The constraint in (3.13b) restricts the E2Edelay of streaming data be bounded above by M slots. The constraints in (3.13c) and (3.13d)together define a feasible single-unicast flow, meeting link bandwidth constraints and flowconservation constraints.

Lemma 9. Suppose M is an arbitrary integer. Given any MPA (or MAA, MMD) instance,suppose R is an arbitrary throughput satisfying R ∈ [Rl, Ru] and D/R ∈ Z+. The problem ofwhether a feasible periodically repeated solution f exists with M(f) ≤ M and T (f) = D/Ris theoretically feasible if and only if the optimal value of the linear program (3.13) is lowerbounded by D.

47

Proof. See our Appendix A.11.

In order to figure out the minimal integer M∗ ∈ [0,MU ] where the associated linear pro-gram (3.13) generates a feasible flow with a value lower bounded by D, we can use a binarysearch scheme according to Lem. 9 (see Algorithm 3). It is clear that such a M∗ is exactlytheMR. We remark that MU used to construct Gexp following Algorithm 2 must exist, sinceit holds that MU ≤ |V | · (dmax +D/Rl) with dmax = maxe∈E de. This is because that in terms

of an arbitrary feasible solution f , we have the following for any p ∈ P and any ~u ∈ ~U :(i) considering that p is a simple path, the aggregate delay of traversing all links on p isbounded above by |V | · dmax, and (ii) considering Lem. 3, the aggregate data-holding delayat all nodes on p is bounded above by |V | ·D/Rl.

Algorithm 3 Obtain MR and the associated periodically repeated solution

1: input: G = (V,E), s, r, D, R, MU

2: output: M, f3: procedure4: f = ft = NULL, M = +∞, LB = 0, UB = MU

5: Construct Gexp following Algorithm 2 with (G,MU)6: while LB ≤ UB do7: M = d(LB + UB)/2e8: ft is the optimal solution of linear program (3.13) with input (Gexp, R,D,M, s, r)9: if the objective of ft is lower bounded by D then

10: M = M , f = ft, UB = M − 111: else12: LB = M + 1

13: returnM, f

Lemma 10. We denote the size of the instance of linear program (3.13) by L. Then Algo-rithm 3 has a time complexity of O(|E|3M3

UL logMU).


According to Lem. 10, the time complexity of Algorithm 3 is pseudo-polynomial, because thesize of the expanded network is pseudo-polynomial: considering MU ≤ |V | · (dmax + D/Rl),it is polynomial with the numeric value of dmax and D/Rl, but is exponential with the bitlength of dmax and D/Rl.

3.5.3 Minimizing AoI Optimally in a Pseudo-Polynomial Time

As introduced in the following lemma, it is highly non-trivial to figure out the throughputRp ∈ [Rl, Ru] (resp. Ra ∈ [Rl, Ru]) that minimizes peak AoI (resp. average AoI).

48

Lemma 11. Both ΛRp and ΛR

a are non-convex, non-concave, and non-monotonic with R.

Proof. See Appendix A.13.

In order to minimize the peak AoI (resp. average AoI) optimally, based on the above lemmawe need to enumerate ΛR

p (resp. ΛRa ) for all R ∈ [Rl, Ru], D/R ∈ Z+, and figure out the

AoI-optimal one. Because we can use Algorithm 3 to achieve ΛRp and ΛR

a , we can solve MPA(resp. MAA) optimally after enumerating all throughputs based on Algorithm 3.

The time complexity of such an enumerating approach is pseudo-polynomial, considering thatthe time complexity of Algorithm 3 is pseudo-polynomial (see Lem. 10), and the number ofpossible throughputs is D/Rl −D/Ru + 1 that is pseudo-polynomial.

Theorem 15. Both MAA and MPA are weakly NP-hard.

Proof. A direct result from Lem. 8 and our proposed pseudo-polynomial-time optimal algo-rithm.

3.6 An Approximation Framework for Minimizing AoI

In the previous Sec. 3.4, we have proved that the AoI achieved by the solution with minimalmaximum delay is also within bounded gaps as compared to the minimal AoI. Thus it isnatural to use solutions of minimizing maximum delay as solutions to MAA and MPA withbounded optimality loss. Note that as discussed in Sec. 3.2, our AoI minimization problems,i.e., MAA and MPA, cover existing maximum delay minimization problems, i.e., the min-max-delay flow problem and the quickest flow problem, as special cases. Now we design aframework that can construct approximate solutions to MAA and MPA in a polynomial time,by leveraging any existing polynomial-time approximation algorithm of the min-max-delayflow problem.

It is clear that any feasible solution f to MAA and MPA with a throughput of R sends Damount of data to G every D/R time slots, with the link bandwidth constraints satisfied. Incomparison, any feasible solution f to the min-max-delay flow problem (detailed definitionof the min-max-delay flow problem refers to [64]) with a throughput of R sends R amountof data to G at each time slot, with the link bandwidth constraints satisfied. Comparing fwith f , it is clear that f is a special case of f , as f can send D amount of data to G everyD/R time slots under link bandwidth constraints.

Suppose MPA(Rl, Ru, D) (resp. MAA(Rl, Ru, D)) is a feasible instance of MPA (resp. MAA)with throughput requirements Rl and Ru, and suppose MMD1(R) is the corresponding min-max-delay flow problem instance with a throughput requirement of R. The following lemmaholds.

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Lemma 12. Given any MPA(Rl, Ru, D) (resp. MAA(Rl, Ru, D)), suppose R ∈ [Rl, Ru], D/R ∈Z+ is an arbitrary feasible throughput. Then it holds that MMD1(R) is feasible. Further-more, if f(R) is feasible to MMD1(R), it must also be a feasible periodically repeated solutionto MPA(Rl, Ru, D) (resp. MAA(Rl, Ru, D)) where we have the following

M(f(R)) = M(f(R)) +D/R− 1.

M(f) in the above equation is the maximum delay of f in MMD1(R).


From Lem. 12 we learn that if a solution is feasible to the min-max-delay flow problem, it isfeasible to the corresponding MAA and MPA, achieving the same throughput. But note thatthe AoI achieved by such a feasible solution can be suboptimal. This is because a solutionto the min-max-delay flow problem always sends R amount of data to G at each time slot,a special case of feasible solutions to MAA and MPA both of which only require D amountof data to be sent every D/R time slots.

Now in the following theorem, we prove that (i) Rl must be feasible to MPA(Rl, Ru, D) (resp.MAA(Rl, Ru, D)), implying that solving MMD1(Rl) must generate a feasible solution toMPA(Rl, Ru, D) (resp. MAA(Rl, Ru, D)) based on Lem. 12; (ii) any approximate solution toMMD1(Rl) must also be an approximate solution to MPA(Rl, Ru, D) (resp. MAA(Rl, Ru, D)).In the following we denote ALG-MMD1(α) as an α-approximation algorithm of the min-max-delay flow problem.

Theorem 16. Given any MPA(Rl, Ru, D) and MAA(Rl, Ru, D) with D/Rl ∈ Z+, D/Ru ∈Z+. We must get an α-approximate solution fα(Rl) to MMD1(Rl), if we use ALG-MMD1(α)to solve MMD1(Rl). Furthermore, fα(Rl) must be an approximate solution to MPA(Rl, Ru, D)and MAA(Rl, Ru, D), with an approximation ratio of (α + c) where c is defined as follows

c =

2 ·Ru/Rl, for MPA(Rl, Ru, D),

1.5 ·Ru/Rl, for MAA(Rl, Ru, D).(3.14)


Based on Thm. 16, we develop a framework that can solve MAA and MPA approximately ina polynomial time, by leveraging any polynomial-time approximation algorithm of the min-max-delay flow problem. If the approximation ratio of the leveraged algorithm is α, thenthe approximation ratio of our framework is a constant depending on α, Rl, and Ru. Weremark that there exist approximation algorithms which can be adopted by our framework,e.g., the (1 + ε)-approximation algorithm (also known as an FPTAS) from [64].

50

a1,1 a1,2a1

a3

a2

Complete Graph Grid Graph

a1,3 a1,4

a2,1 a2,2 a2,3 a2,4

a3,1 a3,2 a3,3 a3,4

a4,1 a4,2 a4,3 a4,4a6

a4

a5

Figure 3.3: Two simulated network topologies in our AoI study.


We carry out extensive simulations in this section, including (i) simulating undirected net-works with well-known topologies (see Fig. 3.3), and (ii) simulating undirected networkswith randomly generated topologies by common-used graph generation models. We treateach undirected link as two independently operated links with opposite directions but samebandwidths and same delays. We randomly generate link bandwidth from 10, 20, 30, 40, 50,and link delay from 1, 2, 3, 4, 5. We consider two differentD withD1 = 5·D andD2 = 10·D,where D is the maximum amount of data that can be sent assuming that the task activationperiod is one, for each simulation instance. We remark that based on our Lem. 12, this Dis also the maximum achieved throughput. Therefore, we observe that 5 (resp. 10) is theminimal feasible task activation period under D = D1 (resp. with D = D2). In each sim-ulation, we consider ten different task activation periods, where T (f) ∈ 5, 6, ..., 14 (resp.T (f) ∈ 10, 11, ..., 19) under D = D1 (resp. with D = D2). We use the FPTAS from [64]as the ALG-MMD1(α) leveraged by our approximation framework, and we set α = 2. Weimplement our simulations in C++, and run them on a laptop with 8 GB memory and aCore i5 (2.40 GHz) processor. We use CPLEX [38] to solve linear programs.

3.7.1 Simulations on Known Topologies

As shown in Fig. 3.3, we first consider two different networks: (i) one is a complete networkwith |V | = 6 and |E| = 15, representing a fully-connected structure; (ii) the other is a gridnetwork with |V | = 16 and |E| = 24, representing a distributed structure. We set s = a1,r = a6 for the complete network, and set s = a1,1, r = a4,4 for the grid network.

Fig. 3.4(a) (resp. Fig. 3.4(b)) shows the simulated AoI of one instance on the completenetwork with D = D1 (resp. on the grid network with D = D2), where Λp(OPT) (resp.Λa(OPT)) is ΛR

p (resp. ΛRa ) that is the peak AoI (resp. average AoI) given by Algorithm 3

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(a) Complete network, assuming D = D1. (b) Grid network, assuming D = D2.

Figure 3.4: Simulated AoI of two instances on networks with known topologies.

achieving a throughput of R, Λp(AP) (resp. Λa(AP)) is the peak AoI (resp. average AoI)given by ALG-MMD1(2) achieving a throughput of R.

From Fig. 3.4(a), empirically we verify our Lem. 7 and 11: (i) the task activation period of6 which minimizes peak AoI is different from that of 8 which minimizes average AoI, and (ii)ΛRp (resp. ΛR

a ) is non-concave, non-convex, and non-monotonic with R.

Next we use the complete network to simulate 100 instances with D = D2 (resp. use thegrid network to simulate 100 instances with D = D1). Fig. 3.5(a) (resp. Fig. 3.5(b)) showsthe simulated AoI in average. From the figure we observe that (i) empirically ΛR

p and ΛRa

are “almost” increasing with the throughput R. Note that for an instance of MAA (resp.MPA), the average AoI (resp. peak AoI) of our approximation framework is the Λa(AP) (resp.Λp(AP)) achieving the smallest throughput (thus the largest task activation period), whilethe average AoI (resp. peak AoI) given by our optimal algorithm is the smallest average AoI(resp. peak AoI) among those achieved by all possible throughputs (thus all possible taskactivation periods). Therefore, (ii) empirically our optimal algorithm obtains a 3.8% peak AoIreduction (resp. 3.2% average AoI reduction) as compared to our approximation framework,when the number of possible throughputs (thus the range of task activation period) of oneproblem instance increases by 1. However, (iii) ALG-MMD1(2) (resp. of Algorithm 3) has anaverage running time of 0.06s (resp. 0.10s) for each throughput. Therefore for one probleminstance, our approximation framework has a running time of 0.06s independent to thenumber of possible throughputs (directly run ALG-MMD1(2) with the smallest throughputrequirement), while the running time of our optimal algorithm increases by 0.10s when thenumber of possible throughputs increases by 1 (enumerate AoIs achieved by all possiblethroughputs to figure out the optimal).

52

(a) Complete network, assuming D = D2. (b) Grid network, assuming D = D1.

Figure 3.5: Simulated AoI of 200 instances on networks with known topologies.

(a) Instances with D = D1. (b) Instances with D = D2.

Figure 3.6: Simulated AoI of 200 instances on networks with randomly generated topologies.

3.7.2 Simulations on Randomly Generated Topologies

We now simulate networks with randomly generated topologies by SNAP [51]. We generatenine networks, where each three of them respectively follow the Watts-Strogatz model, theErdos-Renyi model, and the Copying model. We set |V | = 20 and |E| = 50. We use defaultvalues for all parameters of the tool SNAP.

We simulate 100 instances respectively with D = D1 and with D = D2, for each of the ninetopologies, where we randomly select the sender and the receiver in each simulated instance.Fig. 3.6 gives the corresponding results, which are very similar to the results on networkswith known topologies as shown in Fig. 3.5. ALG-MMD1(2) (resp. of Algorithm 3) has anaverage running time of 0.05s (resp. 0.09s) for instances with D = D1, and 0.06s (resp.0.12s) for instances with D = D2, for each throughput.

In summary, for the AoI minimization problems MPA and MAA, in this section we empirically

53

observe that (i) our optimal algorithm obtains more than 3% AoI reduction as compared toour approximation framework each time the range of task activation period increases by 1;but (ii) our approximation framework has a running time of 0.06s that is independent to therange of task activation period, while the running time of our optimal algorithm increasesby 0.12s if the range of task activation period increases by 1.

3.8 Chapter Summary

We consider a scenario where in a multi-hop network a sender requires to use multiple pathsto send a batch of data to a receiver periodically. We study two problems, with one mini-mizing peak AoI and the other minimizing average AoI, by jointly optimizing (i) multi-pathrouting strategy subject to link bandwidth constraints, and (ii) the throughput subject to amaximum and a minimum throughput requirement. Our AoI problems differ from existingones in that we consider a multi-path communication setting and assumes a batch-based AoIdefinition, while they do not. In this chapter we prove that our problems are weakly NP-hard,since we design an optimal algorithm for solving our problems in a pseudo-polynomial time.We further prove that minimizing maximum delay is “largely” consistent with minimizingAoI, since the solution minimizing maximum delay must be approximate to the problem ofminimizing AoI. This critical observation motivates us to develop a framework which con-structs an approximate solution of minimizing AoI in a polynomial time, by leveraging anyexisting polynomial-time approximation algorithm of minimizing maximum delay. Finally,we carry out extensive simulations with different network topologies to empirically evaluateour approaches.

Chapter 4

Maximize Network Utility withThroughput Requirements andMaximum Delay Constraints

4.1 Introduction

In this chapter we consider a multiple-unicast network communication scenario where eachunicast denotes an user (sender-receiver pair). In a multi-hop network each user requiresa sender to use multiple paths to stream a flow to a receiver. We study the problem ofmaximizing aggregate user utilities under user throughput requirements as well as maximumdelay constraints. A user’s utility is assumed to be a concave function of the experiencedmaximum delay or the achieved throughput. The maximum delay is the maximum pathdelay of paths streaming a positive rate of one user.

We observe that nowadays time-sensitive networking applications, e.g., video conferenc-ing [12, 29, 58], are becoming increasingly important. Recent reports say that nowadaysup to 75% of innovators utilize video collaboration1, and up to 51 million users attend We-bEx meetings per month2. ITU [39] suggests that for highly interactive services like the videoconferencing, an E2E delay within 150ms can provide a service of transparent interactivityfor users, while an E2E delays greater than 400ms are unacceptable. In order to provide lowdelay services for all users, we remark that user’s maximum delay is a critical concern, sinceit upper bounds user’s E2E delay by its definition.

We consider a delay model where the experienced delay of passing a link is a constant ifthe aggregate link traffic is no greater than the capacity; otherwise, it is arbitrarily large.This model fits many applications, particularly video conferencing. To be specific, both

1Cisco, http://www.cisco.com/2WebEx, https://blog.webex.com/

54

55

Table 4.1: Compare existing studies with our NUM work.Optimization Objective Optimization Constraints Setting

Aggregate Throughput-Based Utilities

Aggregate Maximum-Delay-Based Utilities

ThroughputRequirements

Maximum DelayConstraints

Multiple-Unicast

Many, e.g., [45, 60,67,89] 3 7 3 7 3

[14, 15,54,64,97] 7 3∗ 3 7 7

[10, 96] 3∗∗ 7 7 3 3

Out Work 3 3 3 3 3

Note. ∗: Studies [14,15,54,64,97] minimize maximum delay, a special case of optimizing aggregatemaximum-delay-based utilities; ∗∗: Studies [10,96] maximize throughput, a special case of optimizing

aggregate throughput-based utilities.

Google [40] and Microsoft [33] say that the link bandwidth is shared for different applica-tions for most realistic inter-datacenter networks, with over-provisioned link capacities: (i)nowadays various applications including delay-sensitive ones (e.g., video conferencing) anddelay-insensitive ones (e.g., data maintenance) use inter-datacenter networks to send theirtraffic. Link capacity is often reserved separately for different kinds of applications; (ii)cloud providers typically reserve link capacities separately for those applications based ontheir characteristics, and over-provision the aggregate capacity of a link by 2 − 3 times. Insuch a setting, an application will experience a constant propagation delay and experienceno queuing delay when sending traffic to a link, if the amount of traffic is no greater thanthe reserved capacity; otherwise, it will experience a large queuing delay. These observationsjustify our delay model.

We summarize related studies in Tab. 4.1 and give the detailed discussion later in Sec. 4.2.In conclusion, our Network Utility Maximization (NUM) problem covers existing prob-lems which either maximize the maximum-delay-constrained throughput or minimize thethroughput-constrained maximum delay as special cases. We are the first to study the fun-damental NUM problem subject to user throughput requirements as well as maximum delayconstraints, where a user’s utility is a concave function either with the experienced maximumdelay or with the achieved throughput. Our specific contributions are as follows.

We prove it is impossible either (i) to generate a feasible solution with all the constraintssatisfied, or (ii) to figure out an optimal solution after we relax maximum delay constraintsor throughput requirements, in a polynomial time, unless P = NP. This is mainly due tothe need of simultaneously considering user throughput requirements and maximum delayconstraints.

We develop an algorithm named PASS (Polynomial-time Algorithm Supporting utility-maximalflows Subject to throughput/delay constraints). The design of PASS uses a novel techniquethat relates the non-convex maximum delay to the convex average delay. It suggests a newavenue for optimizing multi-path network communications with the maximum delay takeninto account.

We characterize conditions under which PASS solves our problem in a polynomial time,and provides (i) a problem-dependent approximation ratio, at a cost of violating either the

56

throughput requirements or the maximum delay constraints by a problem-dependent ratio,or (ii) a constant approximation ratio, at a cost of violating both maximum delay constraintsand throughput requirements also by constant ratios.

We show that the characterized conditions are met in many popular application scenarios,e.g., maximizing throughput-based utility under maximum delay constraints and minimiz-ing maximum delay under throughput requirements, where we can thus use PASS to solvethose problems approximately with performance guarantee. We simulate video-conferencingapplications across the Amazon EC2 datacenters to evaluate PASS empirically. The utilityachieved by PASS can be up to 100% more than that achieved by conceivable baselines,with the throughput requirements satisfied but the maximum delay constraints violated.We observe that the delay violation of PASS is acceptable for practical video conferencingapplications.


There exist many NUM studies in the literature with throughput concerns, e.g., [45,60,67,89],but less of them take the maximum delay into account. This is because that maximum delayis non-convex, and thus most maximum-delay-sensitive problems are NP-hard [64].

Misra et al. [64] study a single-unicast network flow problem of minimizing maximum delayunder a throughput requirement, and design an FPTAS. Zhang et al. [97] design an FPTASto minimize single-unicast maximum delay further with reliability concerns. Both FPTASesuse binary search to solve flow problems iteratively in time-expanded networks. We notethat this technique only works in the single-unicast scenario. It is unclear of generalizing thetechnique to the multiple-unicast setting with multiple sender-receiver pairs.

Cao et al. [10] design an FPTAS for a multiple-unicast network flow problem of maximiz-ing throughputs under maximum delay constraints. Yu et al. [96] develop FPTASes forother throughput maximization problems for IoT applications. Similar to those from [64,97],FPTASes of [10, 96] is very time-consuming, as they use time-expanded networks to solveproblems iteratively. Furthermore, FPTASes of [10, 96] leverages the idea of primal-dual al-gorithm, where their primal/dual problems are formulated as linear programs. It is unclear ofgeneralizing their technique to the general scenario of maximizing the network utility, wherethe utility is a concave function with throughput. In the literature some other studies alsoexist and take the maximum delay into account. But they only design heuristic approacheswithout performance guarantee, e.g., the study from [58].

Different from the constant link delay model used in in [10,58,64,96,97], some studies considera more general traffic-dependent link delay model. For example, under the traffic-dependentdelay model, Correa et al. [14, 15] and Liu et al. [54] develop different approximation algo-rithms to minimize maximum delay. Our study assumes a constant link delay model, sameas those in [10, 58, 64, 96, 97], but different from those in [14, 15, 54]. Note that maximum-

57

delay-aware problems assuming different delay models are fundamentally different, as inthe single-unicast setting, an FPTAS exists to minimize maximum delay with the constantmodel [64]; but it has been proven that no FPTAS exists to minimize maximum delay withthe traffic-dependent model [15], unless P = NP.

Overall for existing network communication studies that consider both throughput and max-imum delay, in order to design algorithms with performance guarantee, they leverage a tech-nique of using expanded networks to solve flow problems iteratively, whose time complexityis impractically large (e.g., at least O(|E|3|V |4L) to minimize the maximum delay in thesingle-unicast setting [64]). It is not clear how to generalize it to our NUM problem underthe multiple-unicast setting. In contrast in this chapter, we design an approximation al-gorithm for maximizing aggregate user utilities under maximum delay constraints and userthroughput requirements, relying on a novel technique that relates the non-convex max-imum delay to the convex average delay. The time complexity of our approach is small(e.g., O(|E|3L) to minimize the maximum delay in the single-unicast setting (see Thm. 18presented later)).

4.3 Preliminary

4.3.1 System Model

We define G , (V,E) to be an arbitrary multi-hop network. We assume |E| to be thenumber of links and |V | to be the number of nodes. Each link e ∈ E has a delay de ≥ 0and a capacity ce ≥ 0. If the rate of streaming data to a link e ∈ E is no greater than ce,the data will experience a delay of de to pass e; otherwise, the delay is +∞. We consider amultiple-unicast (multiple-user) setting, where each user i requires a sender si ∈ V to usemultiple paths to send a flow to a receiver ti ∈ V \si.

We define Pi as the set of si − ti paths, and P , ∪Ki=1Pi. The delay dp of a path p ∈ P is

dp ,∑

e∈E:e∈p

de.

We define f , fi, i = 1, 2, ..., K as a multiple-unicast flow, where each fi , xp : xp ≥0, p ∈ Pi is a single-unicast flow of the user i. For each fi we define

xei ,∑

p∈Pi:e∈p

xp

as the aggregate rate of e ∈ E of user i. Similarly, we define xe as the aggregate link rate ofe ∈ E of all unicasts, and

xe ,K∑i=1

xei =∑

p∈P :e∈p

xp.

58

We define the throughput (flow rate) of fi by |fi|,

|fi| ,∑p∈Pi

xp =∑

e∈Out(si)

xei =∑

e∈In(ti)

xei ,

where the set of incoming (resp. outgoing) links of v is denoted by In(v) (resp. Out(v)). Themaximum delay of the single-unicast flow fi is defined as

M(fi) , maxp∈Pi:xp>0

dp,

i.e., the maximum delay experienced by a flow unit from si to ti. The total delay of fi is

T (fi) ,∑p∈Pi

(xp · dp) =∑e∈E

(xei · de).

The average delay of fi is A(fi) , T (fi)/|fi|. Without loss of generality, we let A(fi) = 0 if|fi| = 0.

For each single-unicast flow fi, (i) we denote its maximum-delay-based utility as−Udi (M(fi)),where Udi (M(fi)) penalizes fi by the maximum delay; (ii) we denote its throughput-basedutility as U ti (|fi|), which rewards fi by the throughput.


In this chapter we define a problem of Maximizing Utilities under maximum Delay constraintsand Throughput requirements (MUDT) as follows,

(MUDT) : obj: either maxK∑i=1

U ti (|fi|), (4.1a)

or max −K∑i=1

Udi (M(fi)), (4.1b)

s.t. |fi| ≥ Ri, ∀i = 1, 2, ..., K, (4.1c)

M(fi) ≤ Di, ∀i = 1, 2, ..., K, (4.1d)

f = f1, f2, ..., fK ∈ X , (4.1e)

where X defines a feasible multiple-unicast flow f meeting link capacity constrains as wellas flow conservation constraints, i.e.,

X ,

∑e∈Out(si)

xei =∑

e∈In(ti)

xei = |fi|, ∀1 ≤ i ≤ K,

59

∑e∈Out(v)

xei =∑e∈In(v)

xei , ∀v ∈ V \si, ti, ∀1 ≤ i ≤ K,

K∑i=1

xei ≤ ce,∀e ∈ E, vars: xei ≥ 0,∀e ∈ E,∀1 ≤ i ≤ K

.

In the formula in (4.1), the objective (4.1b) (resp. (4.1a)) maximizes user aggregate maximum-delay-based utilities (resp. throughput-based utilities), constraints (4.1d) are maximumdelay constraints of each user i, i = 1, 2, ..., K, and constraints (4.1c) are throughput require-ments of each user i, i = 1, 2, ..., K.

Finally in this section, we remark that it is highly challenging to design algorithms whichcan solve MUDT approximately in a polynomial time, even after we relax constraints, dueto the following theorem.

Theorem 17. For MUDT, it is NP-complete (i) to figure out the optimal solution after werelax either the user throughput requirements or the maximum delay constraints, or (ii) tofigure out a feasible solution with all the constraints satisfied.


4.4 An Algorithm PASS for Maximizing Aggregate User

Utilities

We now develop the polynomial-time algorithm PASS. Under derived conditions, it cangenerate approximate solutions to MUDT meeting relaxed constraints.

4.4.1 Algorithmic Structure of PASS

We note that it is challenging to solve MUDT because of the non-convex maximum delays.The fundamental idea of PASS is to replace the non-convex maximum delays of MUDTby the convex average delays, and then solve the average-delay-based problem instead, inorder to solve MUDT approximately. (i) The average-delay-aware counterpart of the MUDTmaximizing throughput-based utilities is formulated below and denoted as MUAT-T,

(MUAT-T) : obj: maxK∑i=1

U ti (|fi|), (4.2a)

s.t. |fi| ≥ Ri, ∀i = 1, 2, ..., K, (4.2b)

T (fi) ≤ Di · |fi|, ∀i = 1, 2, ..., K, (4.2c)

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Algorithm 4 Algorithm PASS

1: input: Problem (4.1), ε ∈ (0, 1)2: output: f = fi, i = 1, 2, ..., K3: procedure4: Formulate either problem (4.2) or problem (4.3) based on the input problem (4.1)5: Get the optimal solution f = fi, i = 1, 2, ..., K to the formulated problem6: xdeletei = ε · |fi|,∀i = 1, 2, ..., K7: for i = 1, 2, ..., K do8: while xdeletei > 0 do9: Find the slowest flow-carrying path pi ∈ Pi

10: if xpi > xdeletei then11: xpi = xpi − xdeletei , xdeletei = 012: else13: xdeletei = xdeletei − xpi , xpi = 0

14: return the remaining flow f = fi, i = 1, 2, ..., K

f = f1, f2, ..., fK ∈ X . (4.2d)

(ii) The average-delay-aware counterpart of the MUDT maximizing delay-based utilities isformulated below and denoted as MUAT-M,

(MUAT-M) : obj: max −K∑i=1

Udi(T (fi)

Ri

), (4.3a)

s.t. |fi| = Ri, ∀i = 1, 2, ..., K, (4.3b)

T (fi) ≤ Di ·Ri, ∀i = 1, 2, ..., K, (4.3c)

f = f1, f2, ..., fK ∈ X . (4.3d)

Algorithm 4 presents our proposed algortihm PASS. It first figures out the optimal solutionf = fi, i = 1, 2, ..., K (line 5) to the average-delay-aware counterpart of MUDT. Next foreach unicast i = 1, 2, ..., K, it deletes a flow rate of ε · |fi| from slowest paths that carrypositive flow rates of fi (line 8). The remaining flow is the solution of PASS.

4.4.2 PASS Solves MUDT Approximately, Violating Constraintsby Constant Ratios

Lemma 13. Given an arbitrary ε ∈ (0, 1), let us assume f = fi, i = 1, 2, ..., K to bethe solution of line 5, and f = fi, i = 1, 2, ..., K to be the solution of line 14, both ofAlgorithm 4. We have the following

ε · M(fi) ≤ A(fi), ∀i = 1, 2, ..., K. (4.4)

61

Proof. Proof is similar to our Lem. 1, and is skipped.

Lem. 13 relates the non-convex maximum delay of each unicast with the convex average delayof the same unicast. With Lem. 13, we can prove that PASS can solve MUDT approximatelyin a polynomial time, if utility functions meet certain conditions.

Theorem 18. Given an arbitrary ε ∈ (0, 1), now we use PASS to solve a feasible instanceof problem (4.1). If the problem satisfies all conditions below

1. for each i = 1, 2, ..., K, for an arbitrary a ≥ 0, U ti (a) is concave, non-negative, andnon-decreasing with a, Udi (a) is convex, non-negative, and non-decreasing with a,

2. for an arbitrary a ≥ 0, we have the following given any σ ≥ 1

Udi (σ · a) ≤ σ · Udi (a), ∀i = 1, 2, ..., K,

then PASS must give a solution f = fi, i = 1, ..., K in a polynomial time, with the following

∣∣fi∣∣ ≥ (1− ε) ·Ri, ∀i = 1, 2, ..., K, (4.5a)

M(fi)≤ Di/ε, ∀i = 1, 2, ..., K, (4.5b)

f = f1, f2, ..., fK ∈ X . (4.5c)

We denote the optimal solution to problem (4.1) by f ∗ = f ∗i , i = 1, 2, ..., K. We have thefollowing if (4.1a) is the objective

K∑i=1

U ti(∣∣fi∣∣) ≥ (1− ε) ·

K∑i=1

U ti (|f ∗i |) . (4.6)

The following holds if (4.1b) is the objective

K∑i=1

Udi(M(fi))≤ 1

ε·K∑i=1

Udi (M (f ∗i )) . (4.7)


According to the above theorem, PASS solves our problem MUDT with a constant approxi-mation ratio, at the cost of (i) violating maximum delay constraints by a constant ratio of1/ε, and (ii) violating throughput requirements by a constant ratio of (1 − ε). By slightlymodifying PASS, in the following sections we respectively develop (i) PASS-M to solve MUDTapproximately, with the maximum delay constraints satisfied, and (ii) PASS-T to solve MUDTapproximately, with the throughput requirements satisfied.

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Algorithm 5 PASS-M: Modify PASS to Strictly Satisfy Maximum Delay Constraints

1: input: Problem (4.1)2: output: f = fi, i = 1, 2, ..., K3: procedure4: Solve the average-delay-aware counterpart of problem (4.1), and get the solution

f = fi, i = 1, 2, ..., K5: for i = 1, 2, ..., K do6: whileM(fi) > Di do7: Find the slowest flow-carrying path pi ∈ Pi8: Let xpi = 09: return the remaining flow f = fi, i = 1, 2, ..., K

4.4.3 PASS-M Solves MUDT Approximately, Violating Through-put Requirements by Problem-Dependent Ratios

PASS-M is presented in Algorithm 5. PASS-M first solves the average-delay-aware counter-part of MUDT, and then deletes rate from the slowest paths that carry positive rates of fitill the maximum delay of fi is no greater than Di, for each uniast i = 1, 2, ..., K.

Theorem 19. Now we use PASS-M to solve a problem (4.1) which satisfies all the conditionsin Thm. 18. Then PASS-M must give a solution f = fi, i = 1, 2, ..., K in a polynomialtime, with the following held∣∣fi∣∣ ≥ (1− εmax) ·Ri, ∀i = 1, 2, ..., K, (4.8a)

M(fi)≤ Di, ∀i = 1, 2, ..., K, (4.8b)

f = f1, f2, ..., fK ∈ X , (4.8c)

where εmax is

εmax = max1≤i≤K

(∣∣∣fi∣∣∣− ∣∣fi∣∣) / ∣∣∣fi∣∣∣ ,where f = fi, i = 1, 2, ..., K is the optimal solution to the problem in line 4 of Algorithm 5.We denote the optimal solution to problem (4.1) as f ∗ = f ∗i , i = 1, 2, ..., K. We have thefollowing if (4.1a) is the objective,

K∑i=1

U ti(∣∣fi∣∣) ≥ (1− εmax) ·

K∑i=1

U ti (|f ∗i |) . (4.9)

The following holds if (4.1b) is the objective,

K∑i=1

Udi(M(fi))≤ 1

εmin

·K∑i=1

Udi (M (f ∗i )) , (4.10)

63

where εmin is

εmin = min1≤i≤K

(∣∣∣fi∣∣∣− ∣∣fi∣∣) / ∣∣∣fi∣∣∣ .Proof. Refer to our Appendix A.18.

Comparing Thm. 18 of PASS with Thm. 19 of PASS-M, (i) the solution of PASS has a con-stant approximation ratio, after we relax both maximum delay constraints and throughputrequirements by constant ratios, while (ii) the solution of PASS-M has a problem-dependentapproximation ratio, after we relax throughput requirements by a problem-dependent ratio.

4.4.4 PASS-T Solves MUDT Approximately, Violating Delay Con-straints by Problem-Dependent Ratios

Our PASS-T is introduced as follows

B PASS-T: directly solve the average-delay-aware counterpart of problem (4.1) and deleteno flow rate afterwards.

Theorem 20. Now we use PASS-T to solve a problem (4.1) which satisfies all the conditionsin Thm. 18. Then PASS-T must give a solution f = fi, i = 1, 2, ..., K in a polynomial time,with the following held assuming that g = g1, g2, ..., gK is the solution if we use PASS tosolve the problem with an ε ∈ (0, 1)∣∣fi∣∣ ≥ Ri, ∀i = 1, 2, ..., K, (4.11a)

M(fi)≤ λ

ε·Di, ∀i = 1, 2, ..., K, (4.11b)

f = f1, f2, ..., fK ∈ X , (4.11c)

where λ is

λ = max

1, max

1≤i≤K

M(fi)/M(gi)

.

We denote the optimal solution to problem (4.1) as f ∗ = f ∗i , i = 1, 2, ..., K. We have thefollowing if (4.1a) is the objective,

K∑i=1

U ti(∣∣fi∣∣) ≥ K∑

i=1

U ti (|f ∗i |) . (4.12)

The following holds if (4.1b) is the objective,

K∑i=1

Udi(M(fi))≤ λ

ε·K∑i=1

Udi (M (f ∗i )) . (4.13)

64


Comparing Thm. 18 of PASS with Thm. 20 of PASS-T, (i) the solution of PASS has a con-stant approximation ratio, after we relax both maximum delay constraints and throughputrequirements by constant ratios, while (ii) the solution of PASS-T has a problem-dependentapproximation ratio, after we relax maximum delay constraints by a problem-dependentratio.

4.4.5 Some Other Important Maximum-Delay-Aware Problems

As shown in the formulation in (4.1), MUDT maximizes aggregate user utilities. As a com-parison, the following gives another two important objectives

max min1≤i≤K

U ti (|fi|)

, (4.14a)

max min1≤i≤K

−Udi (M(fi))

. (4.14b)

Following the proof to Thm. 18, 19, and 20, we observe that PASS, PASS-M, and PASS-T all are polynomial-time approximation algorithms to the problem with an objective ofeither (4.14a) or (4.14b) under constrains (4.1c), (4.1d), and (4.1e), as long as the conditionsin Thm. 18 are met. Thus in this chapter we suggest a new technique for optimizing multi-path network communications that consider both throughputs and maximum delays.

4.5 Popular Network Communication Settings Sensi-

tive Both to Throughput and to Maximum Delay

We introduce two popular network communication scenarios both of which are sensitive tothe maximum delay and to the throughput in this section. Although corresponding problemsare both NP-hard, we observe that MUDT cover both of them as special cases. Since theconditions in Thm. 18 are satisfied by both problems, we can use PASS, PASS-M, and PASS-Tto solve them with performance guarantee.

4.5.1 Minimizing Maximum Delay under Throughput Require-ments

The Throughput-Constrained maximum Delay Minimization problem (TCDM) has the fol-lowing formulation

(TCDM) : minK∑i=1

(wi · M(fi)) (4.15a)

65

s.t. |fi| ≥ Ri, ∀i = 1, 2, ..., K, (4.15b)

f = f1, f2, ..., fK ∈ X , (4.15c)

where wi ≥ 0 a non-negative weight associated with the maximum delay of each unicasti = 1, 2, ..., K.

TCDM is proven to be NP-hard [64]. Similar problems of minimizing maximum delay havebeen studied in [14, 15, 54, 64, 97]. It is easy to verify that TCDM meets our characterizedconditions in Thm. 18. Thus we can either (i) use PASS to solve it with a constant ap-proximation ratio, after we relax throughput requirements (see Thm. 18), or (ii) use PASS-Tto solve it with a problem-dependent approximation ratio, with throughput requirementssatisfied (see Thm. 20).

4.5.2 Maximizing Throughput-Based Utility under Maximum De-lay Constraints

The maximum-Delay-Constrained throughput-based Utility Maximization problem (DCUM)has the following formulation

(DCUM) : maxK∑i=1

U ti (|fi|) (4.16a)

s.t. M(fi) ≤ Di, ∀i = 1, 2, ..., K, (4.16b)

f = f1, f2, ..., fK ∈ X . (4.16c)

DCUM can be proven to be NP-hard, following a proof similar to that of Appendix of [64].Similar problems of maximizing throughput-based utilities have been studied in [10, 96].We note that it is practically fair to assume concave, non-negative, and non-decreasingthroughput-based utility functions, implying that our conditions of Thm. 18 are met. There-fore, we can either (i) use PASS to solve it with a constant approximation ratio, after we relaxmaximum delay constraints (see Thm. 18), or (ii) use PASS-M to solve it with a problem-dependent approximation ratio, with maximum delay constraints satisfied (see Thm. 19).


By simulating video conferencing traffic over 6 globally distributed Amazon EC2 datacenters,in this section we empirically evaluate our proposed algorithms. The Amazon EC2 networkis a real-world continent-scale inter-datacenter network, with a complete graph topologyshown in Fig. 4.1. We treat each undirected link as two independent directed links withopposite directions, identical capacities, and identical delays. Link capacities and delays

66

Figure 4.1: Simulated network of 6 Amazon EC2 datacenters [58].

Table 4.2: (de, ce) of each link e ∈ E in Amazon EC2 [29,58], where ce is capacity (in Mbps)and de is delay (in ms).

OR VA IR TO SI SPOR N/A (41,82) (86,86) (68,138) (117,74) (104,67)VA - N/A (54,72) (101,41) (127,52) (82,70)IR - - N/A (138,56) (117,44) (120,61)TO - - - N/A (45,166) (151,41)SI - - - - N/A (182,33)SP - - - - - N/A

are set as Tab. 4.2, which is based on practical evaluations from [29, 58]. We assume twounicasts, where s1 is Virginia, t1 is Singapore, s2 is Oregon, and t2 is Tokyo. We implementour simulations in C++, and run them on a laptop with 8 GB memory and a Core i5 (2.40GHz) processor. We solve linear programs using CPLEX [38].

4.6.1 Simulating the Problem of Minimizing Maximum Delay

In this section we use PASS to solve TCDM with formula (4.15), which minimizes maximumdelay under user throughput requirements. We assume w1 = w2 and R1 = R2 = R.

We compare PASS with a conceivable baseline, the optimal solution, and PASS-T respectively:(i) we note that the delay of any path must be an integer due to the integral link delay inour simulations. Therefore, we can obtain the optimal solution that minimizes maximumdelay in an exponential time, by enumerating all possible maximum delays using the time-expanded network. Such an approach is the foundation of the FPTAS [64]; (ii) in order tominimize maximum delay, the conceivable baseline greedily allocates as much rate as possibleto the shortest paths of the unicast i iteratively, till a total rate of Ri is assigned under linkcapacity constraints, for each i = 1, 2, ..., K. We observe that similar approaches have beenused in the literature, e.g., in [22].

First, we vary ε from 1% to 99% with a step of 1%, and set R = 230, where Fig. 4.2(a)gives the simulated results. We observe that (i) the solution of PASS-T is optimal, (ii) the

67

(a) Delay results with ε of PASS, whereR1 = R2 = 230.

(b) Delay results with throughput require-ments, where ε = 3% in PASS.

Figure 4.2: Simulate the problem of minimizing maximum delay.

solution of the baseline is suboptimal, and (iii) the delay of PASS is a step function with ε.

Second, we vary R from 116 to 239 with a unit step. We remark that 239Mbps is thelargest throughput that can be streamed, and 116Mbps is the smallest throughput when thetwo unicasts both need multiple paths to stream the throughput. Considering that a 3%throughput loss is acceptable in video conferencing [92], we set ε = 3%. Fig. 4.2(b) givesthe simulated results. In average, the baseline experiences a maximum delay (402) that is11% more than that experienced by the PASS (359) and the optimal (362). In the worstcase (R ∈ [116, 138]), the baseline experiences a maximum delay that is 40% more than thatexperienced by the PASS and the optimal. Besides, in most instances the solution of PASS-Tis optimal, except for R ∈ [212, 223].

4.6.2 Simulating the Problem of Maximizing Throughput

We then use PASS to solve DCUM with formula (4.16), which maximizes throughput undermaximum delay constraints. We assume U t1(|f1|) = |f1|, U t2(|f2|) = |f2|, and D1 = D2 = D.We compare PASS with a conceivable baseline, the optimal solution, and PASS-M respec-tively. Here both the conceivable baseline and the optimal solution are constructed followingmethods similar to those discussed in Sec. 4.6.1.

We set D = 150, considering that for video conferencing, ITU [39] suggests that (i) if an E2Edelay is within 150ms, it can guarantee a transparent interactivity; (ii) otherwise, it is stillacceptable if within 400ms. Therefore, although the solution of PASS can violate maximumdelay constraint, it may still be useful if the achieved throughput is large.

We vary ε from 1% to 99% with a step of 1%. Fig. 4.3(a) gives the throughput results, andFig. 4.3(b) presents the achieved maximum delay ratio results (i.e., maxM(f1),M(f2)/Dwhere f = f1, f2 is the solution). We observe that both PASS-M and the baseline obtain

68

(a) Throughput results (both baseline andPASS-M obtain the optimal).

(b) Delay ratio comparing the achieved re-sult to the constraint.

Figure 4.3: Simulate the problem of maximizing throughput. We vary ε and assume D1 =D2 = 150.

the optimal throughput with the maximum delay constraints satisfied in our simulations.For ε ≤ 49%, PASS achieves a throughput greater than optimal, but the maximum delayconstraints are violated (e.g., 8% more than D when ε = 49%). For ε ≥ 51%, PASS achievesa throughput smaller than optimal, but the maximum delay constraints are satisfied. It isimpressive that for our algorithm PASS given a small ε, e.g., ε = 1%, it increases the achievedthroughput by more than 90% as compared to the optimal, while the experienced maximumdelay is less than 331ms that is acceptable for video conferencing. Averagely speaking, whenε is decreased by 1% for instances where ε ≤ 49%, as compared to the optimal, we observea 2.0% improvement in the throughput, but with a 2.2% violation in the delay constraints.

4.6.3 Simulating the Problem of Maximizing Network Utility

Finally, we use PASS to solve MUDT with formula (4.1)) which maximize utilities under userthroughput requirements as well as maximum delay constraints. We assume the objectiveis (4.1a) where U ti (|fi|) = wi·|fi|, i = 1, 2. And we assumeR1 = R2 = 80, andD1 = D2 = 150.

We vary w1 (resp. w2) from 1 to 10 with a step of 1, resulting in 100 different simulationinstances. We compare PASS, PASS-M, PASS-T, respectively with the optimal.

Fig. 4.4(a) shows the utility results of different algorithms. Fig. 4.4(b) gives the increment(%) comparing our achieved utility with the optimal. Here we remark that because ouralgorithms PASS, PASS-M, and PASS-T are all under relaxed constraints, their achievedutilities can be larger than the optimal that are under original constraints.

We observe that the utility of PASS-M is near-optimal, while the utilities of PASS andPASS-T are 100% more than optimal. For the first unicast (resp. second unicast), averagelyspeaking, (i) the throughput of PASS is 138 (resp. 302) , both meeting the requirement

69

(a) Utility results of different algorithms,with ε = 3% in PASS.

(b) Utility increment compared to opti-mal, with ε = 3% in PASS.

Figure 4.4: Simulate the problem of maximizing utility, where we set R1 = R2 = 80 andD1 = D2 = 150.

R1 = R2 = 80, and the maximum delay of PASS is 195 (resp. 301), both violating theconstraint D1 = D2 = 150. However, considering that both experienced maximum delays arewithin 400ms, the solution of PASS is acceptable for video conferencing; (ii) the throughputof PASS-M is 71 (resp. 154). It is clear that the first unicast flow violates throughputrequirement; (iii) the maximum delay of PASS-T is 222 (resp. 322), both violating theconstraints but still within the acceptable delay of 400ms.

4.7 Chapter Summary

In this chapter we study a fundamental network utility maximization problem under userthroughput requirements as well as maximum delay constraints. We consider a user’s util-ity to be a concave function of the experienced maximum delay or the achieved throughput.First we argue that it is impossible either (i) to generate an arbitrary solution strictly satisfy-ing all the constraints, or (ii) to construct an optimal solution after we relax user throughputrequirements or maximum delay constraints, in a polynomial time unless P = NP. Next wedevelop an algorithm named PASS. Under realistic conditions, PASS must obtain approxi-mate solutions in a polynomial time, at the cost of violating user throughput requirements ormaximum delay constraints by bounded ratios. We simulate a realistic network connecting 6Amazon EC2 datacenters, serving video-conferencing traffic, in order to empirically evaluatePASS. The design of PASS presents a new technique that relates the non-convex maximumdelay to the convex average delay, suggesting a new avenue for optimizing multi-path networkcommunications that take both throughput and maximum delay into account.

Chapter 5

Energy-Efficient Timely TruckTransportation for Fulfilling MultipleTasks

5.1 Introduction

In the US in 2016, heavy-duty trucks hauls 70.6% of all freight tonnage [1], and collects 676billion dollars in gross freight revenues [1]. If measured against the GDP of countries, thisnumber would rank 19th worldwide. With just 4% of vehicle population, heavy-duty trucksconsume 18% of energy in the whole transportation sector [34]. This alerting observation,as well as another observation that fuel consumption is the largest truck operating cost(26%) [34], makes it highly critical to save fuel for environment-friendly and cost-effectiveheavy-duty truck operation.

In this chapter we consider a common truck operation scenario where in a national highwaynetwork a heavy truck is traveling to fulfill many ordered transportation tasks. We studya timely eco-routing problem of minimizing fuel consumption under task pickup and deliv-ery time window constraints. Nowadays mobile applications like uShip1 and Uber Freight2

provide freight transportation tasks for truck operators, which are often associated withearliest/latest pickup/delivery time requirements. We minimize fuel consumption by jointlyoptimizing path planning, speed planning, and task execution times. Path planning andspeed planning are two well-recognized approaches for saving fuel3, as discussed in the ex-isting studies [18,19].

In addition to path planning and speed planning, we consider a new design space of task

1uShip, https://www.uship.com/2Uber Freight, https://freight.uber.com/3US Department of Energy, https://afdc.energy.gov/data/

70

71

Table 5.1: Compare existing energy-efficient timely truck transportation studies with ours.Studied problem RSP [30, 41,59] PASO [18, 19] [31,32] [9, 79] Our work

Setting Fulfill multiple tasks 7 7 7 7 3

Designspace

Path planning 3 3 7 3 3

Speed planning 7 3 3 7 3

Task execution times optimization 7 7 7 7 3

Constraint Time window constraints 3 3 7 7 3

execution times optimization for saving fuel, which differentiates our study under the multi-task setting from existing ones under the single-task setting. We note that it is necessary tooptimally coordinate execution times for individual tasks by jointly considering their timewindow constraints. To be specific, for individual tasks, a longer execution time budgetcan save fuel due to a bigger design space of path planning and speed planning. However,we cannot allocate an arbitrarily long execution time budget for each task, because it iscrucial to catch time window constraints to avoid penalty of violation. Further, due to thatdifferent tasks can have temporally overlapped time window constraints, although increasingthe execution time budget of one task can save its fuel, overall it may consumes more fuel tofulfill all tasks because of the decreasing execution time budgets of other tasks. In conclusion,task execution times optimization is a must for effectively saving fuel in the multi-task setting.

We remark that optimizing task execution times is challenging, as it is a combinatorialoptimization problem4 since time windows of different tasks can overlap.

We compare our study with existing ones in Tab. 5.1, and give details of related work laterin Sec. 5.2. In summary, we are the first to study the fundamental transportation problemof minimizing fuel consumption for a heavy truck to fulfill multiple tasks subject under taskpickup and delivery time window constraints. Note again that solving our problem requiresto simultaneously optimize speed planning, path planning, and task execution times. Weclaim the following specific contributions.

We show that our problem is NP-hard, and it is non-convex even to optimize the taskexecution times by itself.

We develop an efficient heuristic SPEED (Sub-gradient-based Price-driven Energy-EfficientDelivery), based on dual sub-gradient updates of task execution times. We characterizesufficient conditions under which the solution of SPEED must be optimal, and further derivea performance gap comparing the solution of SPEED with the optimal when the conditionsare not satisfied.

We simulate a truck driving across the US national highway system to evaluate our solu-

4Consider an instance where all time windows of K tasks are [0, T ] with T ∈ Z. If tk ∈ [0, T ] is the assigned

execution time budget for the task k, then we should have∑K

k=1 tk ≤ T to guarantee timely deliveries of alltasks. Now if we restrict that tk ∈ Z, k = 1, 2, ...,K, the number of feasible allocations tk : k = 1, 2, ...,Kwill be O(KT ) that is exponential with the input T .

72

tions. As compared to the fuel consumption achieved by the fastest-/shortest- path baselines(resp. a conceivable approach generalized from the state-of-the-art single-task algorithm),our solutions can reduce the fuel consumption by 22% (resp. 10%). The fuel saving of oursolutions is independent to the number of tasks to be fulfilled.


Restricted Shortest Path (RSP): it requires to find a path with an objective of minimizing pathfuel consumption and a constraint that restricts the path travel time to be upper boundedby a given deadline. RSP is NP-hard [30] with heuristic algorithms [41] and FPTASes [30,59]designed. RSP optimizes path planning to save fuel, while road driving speeds are fixed.In addition, RSP considers single task and thus no task execution times optimization isinvolved. It cannot be directly extended to our multi-task setting where optimizing taskexecution times is challenging and is a must for saving fuel.

PAth selection and Speed Optimization (PASO): it generalizes RSP with speed planning inconsideration. Deng et al. [18, 19] develop both a heuristic and an FPTAS to solve PASO.Similar to RSP, PASO considers single task. Hence its solution cannot be easily extended toour multi-task scenario, as solving the multi-task problem involves addressing a challengingpuzzle of task execution times optimization.

Pickup and Delivery Problem with Time Window (PDPTW): it is an extension of the Travel-ing Salesman Problem (TSP) [50] and Vehicle Routing Problem (VRP) [48]. PDPTW [49,65,77] and its extensions [66,69] minimize fuel consumption for a set of vehicles to timely fulfillmultiple tasks. However, we remark that PDPTW fundamentally differs from our problem:(i) in order to fulfill one task, we optimize path planning and speed planning while PDPTWassumes fixed paths (e.g., shortest paths) and fixed road driving speeds; (ii) the main chal-lenge of PDPTW is to optimize the execution order of tasks, while we assume the executionorder of tasks is given.

Other studies : Sahlholm et al. [75] introduce a method of estimating road grade. Suchgrade information can be used by assistance systems to optimize driving speed to save fuel,as studied in [31, 32]. Note that studies [31, 32] consider the problem of fulfilling one taskwithout time window constraint, assuming that the path is given and thus no path planningis involved. Boriboonsomsin et al. [9] introduce an eco-routing system that figures out thepath with minimal fuel consumption. Scora et al. [79] relate the fuel savings to the traveltime budget. Both studies [9, 79] consider only single task without time window constraint,assuming fixed road driving speeds and thus no speed planning is involved. Alam et al. [4]argue that improved fuel-efficiency can be achieved by maintaining a platoon, motivatingsubsequent studies, e.g., [3, 5], all of which focus on designing control strategies for truckplatooning for saving fuel.

Overall, we are the first to study a truck routing problem of minimizing fuel consumption for

73

(a) System model: an example of a truckdriving in a highway network.

(b) Simulated US national highway sys-tem that is partitioned into 22 regions.

Figure 5.1: A truck timely fulfills multiple tasks in a national highway network.

fulfilling multiple tasks, subject to task pickup and delivery time window constraints. Solvingthe problem requires us to jointly optimize speed planning, path planning, and task executiontimes. As compared to existing single-task studies, task execution times optimization is anew challenging but necessary design space for saving fuel for our problem that is under themulti-task setting.

5.3 Preliminary

5.3.1 System Model

We define a directed graph G , (V,E) as a national highway network. An edge e ∈ Erepresents a road segment, and a node v ∈ V represents a connecting point. A road segmentis assumed to have homogeneous environmental conditions, e.g., grade, that impact the fuelconsumption rate of a truck. We define n , |V | and m , |E|. We denote De > 0 as roaddistance, rle > 0 (resp. rue ≥ rle) as road minimum (resp. maximum) driving speed, andcle > 0 (resp. cue ≥ cle) as road minimum (resp. maximum) fuel consumption, for each roadsegment e ∈ E.

We consider a common truck operation scenario, where one heavy-duty truck travels acrossG to fulfill K tasks denoted by ~τ = τi, i = 1, 2, ..., K, as illustrated in Fig. 5.1(a). Wemake the following assumptions:

1. the truck must fulfill tasks in a specific order, namely the ith task τi must be fulfilledbefore the jth task τj for any 1 ≤ i < j ≤ K, and

2. the truck cannot simultaneously transport cargoes belonging to different tasks.

We define ρi ≥ 0 as the cargoes’ mass of a task τi. In addition, each τi is characterized by asource si ∈ V where the truck can pick up cargoes, a destination di ∈ V to which cargoes

74

Table 5.2: Summary of used notations in our truck transportation study.

G , (V,E)The highway network G with connecting

points V and road segments Ece(·) Fuel consumption function of e ∈ E

rle (resp. rue ) Minimum (resp. maximum) speed of etle (resp. tue ) Minimum (resp. maximum) travel time of e

cle (resp. cue )Minimum (resp. maximum) fuelconsumption of road segment e

te(ρ)The specific travel time t ∈ [tle, t

ue ] that

minimizes ce(·) given a truck load ρτi, i ∈ [1, K] The ith task

si (resp. di)Source node (resp. destination

node) of the ith task

sωi (resp. dωi)Source pickup window (resp. destination

delivery window) of the ith taskρi Weight of cargos (load) of the ith task

α(sωi) (resp.β(sωi))

Earliest (resp. latest) pickuptime of the ith task

α(dωi) (resp.β(dωi))

Earliest (resp. latest) deliverytime of the ith task

σi, i ∈ [1, K + 1] Node si, or equivalently the node di−1

T outi (resp. T in

i )Earliest leaving (resp. latest arrival) time ofσi by jointly considering sωi and dωi−1

are delivered, a pickup time window sωi representing the allowed time for picking up at si,and a delivery time window dωi defining the allowed time for delivering at di. We use α(·)(resp. β(·)) to denote the starting time (resp. ending time) of a time window, i.e.,

sωi , [α(sωi), β(sωi)] , β(sωi) ≥ α(sωi) ≥ 0,

dωi , [α(dωi), β(dωi)] , β(dωi) ≥ α(dωi) ≥ 0.

When the truck picks up the task τi at si, the pickup window sωi restricts that (i) it cannotarrive at si later than the time β(sωi), and (ii) it cannot leave si until the time α(sωi). Thesame assumption holds for the delivery window dωi when the truck delivers the task τi at di.

Many factors can impact the truck fuel consumption [17]. Like studies [18,19], we ignore fuelconsumption caused by acceleration or deceleration of a truck. This is mainly because (i) asdiscussed in [11,25] and proved by [19, Lem. 1], for driving inside any road segment with ho-mogeneous environmental conditions, it is most fuel-economic by following a constant-speed;(ii) although a truck may accelerate or decelerate when switching between different road seg-ments, compared with the length of road segments, the distance of acceleration/decelerationis negligible. Hence we can fairly ignore the fuel consumption from acceleration/decelerationwhen switching between road segments, as compared to that incurred by driving insider aroad segment. Overall, given a road segment, it is reasonable to model the fuel consumptionrate of a heavy-duty truck using a function of the driving speed [18,19] and the cargo mass.

We define fe(re, ρ) :[rle, r

ue

]→ R+ as the fuel consumption rate for the truck to pass e ∈ E

75

with a speed of re and a load of ρ. Same to the assumption made by [18,19] and verified byboth physical laws and simulations using real-world data, we assume fe(re, ρ) to be strictlyconvex with re over the interval

[rle, r

ue

], given a truck load ρ ≥ 0.

With the fuel consumption rate function fe(re, ρ), we can define fuel consumption functionce (te, ρ) as follows

ce (te, ρ) , te · fe(De

te, ρ

), (5.1)

which is the truck fuel consumption of traversing e with a load of ρ and a travel time ofte. Due to the strict convexity of fe(·), following the proof of [18, Lem. 2], given a truckload of ρ we have: (i) ce (te, ρ) is strictly convex over te ∈

[tle, t

ue

], where tle , De/r

ue is the

minimum travel time and tue , De/rle is the maximum travel time, and (ii) there exists a

travel time te(ρ) ∈[tle, t

ue

]such that ce (te, ρ) is first strictly decreasing over

[tle, te(ρ)

]and

then strictly increasing over[te(ρ), tue

]. Hence, to fulfill task τi, the possible travel time in

the optimal solution must belong to the range of[tle, te(ρi)

], for each edge e ∈ E. Without

loss of generality, we assume ce(te(ρi), ρi

)≥ cle and ce

(tle, ρi

)≤ cue .


Without loss of generality, considering any two consecutive tasks τi and τi+1 (1 ≤ i ≤ K−1),we assume that the delivery node of the ith task is the same as the pickup node of the (i+1)th

task, i.e. di = si+1. Similarly, we assume s1 to be the source and dK to be the destinationof the whole trip. With above assumptions, the task sequence ~τ is equivalent to a nodesequence ~σ for the truck to pass:

~σ , σi : σ1 = s1, σK+1 = dK , σi = si = di−1, i = 2, 3, ..., K.

For any σi, because it is the destination of τi−1 as well as the source of τi, it has twotime window constraints, i.e., dωi−1 and sωi. By jointly considering the two time windowconstraints, for σi, the earliest allowed leaving time is in fact maxα(dωi−1), α(sωi), denotedby T out

i ,T out

1 , α(sω1), T outi , max α(dωi−1), α(sωi) , i = 2, 3, ..., K,

indicating that the truck cannot leave σi until it can finish the delivery of task τi−1 and thepickup of task τi. Similarly, the latest allowed arrival time, denoted by T in

i , is

T inK+1 , β(dωK), T in

i , min β(dωi−1), β(sωi) , i = 2, 3, ..., K,

indicating that the truck must arrive at σi before the latest delivery time of task τi−1 andthe latest pickup time of task τi. Overall, all time window constraints now correspond tothe earliest leaving time constraint and the latest arrival time constraint for nodes in ~σ.

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We consider two kinds of design variables: binary variable xei defines a path from σi to σi+1

to fulfill τi,

xei =

1, e ∈ E is on the path to fulfill the task τi;0, otherwise,

and non-negative variable tei defines the travel time for the truck to pass e to fulfill τi.

By vectoring variables as ~xi , xei : e ∈ E and ~ti , tei : e ∈ E, our problem Multi-taskEnergy-Efficient Trucking, denoted by MEET, has the following formulation

obj: min~xi∈Xi,~ti∈Ti

K∑i=1

∑e∈E

xei · ce(tei , ρi) (5.2a)

s.t. ai = max ai−1, Touti +

∑e∈E

xei · tei ≤ T ini+1,

a0 = 0, ∀i = 1, 2, ..., K, (5.2b)

where Ti defines feasible road travel time, i.e.,

Ti ,~ti : tle ≤ tei ≤ te(ρi), ∀e ∈ E

,

and Xi represents the set of paths from σi to σi+1, i.e.,

Xi ,~xi : xei ∈ 0, 1,∀e ∈ E, and

∑e∈out(v)

xei −∑e∈in(v)

xei = 1v=σi − 1v=σi+1,∀v ∈ V,

where in(v) , (u, v) : (u, v) ∈ E (resp. out(v) , (v, u) : (v, u) ∈ E) is the set ofincoming (resp. outgoing) edges of v ∈ V , and 1· is the indicator function.

In the formulation in (5.2), ai is the truck arrival time at σi+1, which should be no laterthan the time T in

i+1. The formula maxai−1, Touti guarantees that the truck cannot leave σi

immediately if it arrives at σi before the time T outi . Objective (5.2a) minimizes total fuel

consumption for fulfilling ~τ .

In this chapter we denote a solution to MEET as

p = p1 ∪ p2 ∪ ... ∪ pK ,

which is a path from σ1 to σK+1 passing all σi ∈ ~σ, with each pi being a simple path fromσi to σi+1, and with each edge e ∈ pi assigned a travel time tei ∈ [tle, te(ρi)].

We present an illustrative example of MEET in Tab. 5.3, assuming the highway network tobe that of Fig. 5.1(a). There are two paths to fulfill τ1 in this example: one is the path 〈1, 3〉with a travel time of 1 and a cost5 of 3, while the other is the path 〈1, 2, 3〉 with a travel timeof 2 and a cost of 2. There are two paths to fulfill τ2: one is the path 〈3, 5〉 with a traveltime of 1 and a cost of 4, while the other is the path 〈3, 4, 5〉 with a travel time of 2 and acost of 2. The optimal solution to MEET in this example is to follow the path 〈1, 3, 4, 5〉,with a total fuel consumption of 5 satisfying time window constraints.

5We interchangeably use fuel consumption and cost.

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Table 5.3: An illustrative example of our studied truck transportation problem MEET basedon Fig. 5.1(a). We set tle = tue = 1 and cle = cue for each e ∈ E.

e (1, 2) (2, 3) (1, 3) (3, 4) (4, 5) (3, 5) (1, 5)ce 1 1 3 1 1 4 8τi τ1 : 1→ 3 τ2 : 3→ 5sωi [0, 2] [0, 3]dωi [0, 2] [0, 3]

5.4 It is Challenging to Solve Our Truck Transporta-

tion Problem

We note that under a single-task setting assuming fixed road driving speeds, MEET requiresthe truck to travel from a source to a destination, with an objective of minimizing fuelconsumption and a constraint which restricts the travel time to be upper bounded by avalue that is the task latest delivery time minus the task earliest pickup time. We observethat this problem is exactly the problem RSP which is NP-hard [30, 59]. Hence MEET isNP-hard.

Theorem 21. MEET is NP-hard.

Proof. MEET is NP-hard because it covers RSP as a special case, and RSP has been provento be NP-hard [30,59].

Now suppose Ci(Ti) is the minimal fuel consumption to fulfill the single task τi with the traveltime upper bounded by a deadline6 of Ti. Then our problem MEET can also be formulatedas follows with an execution time budget Ti assigned to τi for each i = 1, ..., K:


K∑i=1

Ci(Ti) (5.3a)

s.t. ai = max ai−1, Touti + Ti ≤ T in

i+1,

a0 = 0, ∀i = 1, 2, ..., K, (5.3b)∑e∈E

xei · tei ≤ Ti, ∀i = 1, 2, ..., K, (5.3c)

If we can find the optimal assignment Ti, i = 1, 2, ...K, we can run existing single-taskalgorithm (e.g., those from [18,19]) K times independently to obtain a high-quality solution.But it is hard to figure out the optimal assignment, because Ci(Ti) is non-convex with Ti.

6We use “deadline” as a constraint on the maximum travel time for the truck to fulfill a task, while “timewindow” gives the earliest and latest pickup and delivery time at a node.

78

Consider the network in Fig. 5.1(a) as an example. Suppose there is one task where sourceis node 1 and destination is node 5. Edge travel time and fuel consumption are the same asdefined in Tab. 5.3. There are 5 paths from source to destination. By enumerating them, wehave C(T ) = 4 when T = 4 following the path 〈1, 2, 3, 4, 5〉, C(T ) = 5 when T = 3 followingthe path 〈1, 3, 4, 5〉, C(T ) = 7 when T = 2 following the path 〈1, 3, 5〉, and C(T ) = 8 whenT = 1 following the path 〈1, 5〉. Then we observe that the minimal fuel consumption C(T )is neither convex nor concave in T .

5.5 An Efficient Heuristic SPEED for Our Problem MEET

We develop a heuristic named SPEED (Sub-gradient-based Price-driven Energy-Efficient De-livery), which can figure out close-to-optimal solutions to our problem MEET quickly. SPEEDiteratively allocates execution times for individual tasks towards the optimal, by followingthe sub-gradient of the Lagrangian dual relaxation of MEET. We characterize sufficient con-ditions under which SPEED outputs an optimal solution, and further derive a performancegap comparing the solution of SPEED with the optimal when the conditions are not met.

5.5.1 A New Problem Formulation of MEET

We first give a new formulation to MEET in the following, which is characterized by deadlineconstraints imposed on any consecutive tasks τk, τk+1, ..., τr,∀r = 1, 2, ..., K,∀k = 1, 2, ..., r:


K∑i=1

∑e∈E

xei · ce(tei , ρi) (5.4a)

s.t.r∑j=k

∑e∈E

xej · tej ≤ T inr+1 − T out

k ,

∀r = 1, 2, ..., K,∀k = 1, 2, ..., r. (5.4b)

Although the formulation in (5.4) differs from that in (5.2), we prove that they are equivalentto each other below.

Theorem 22. The problems in (5.2) and (5.4) are equivalent in that they share the sameobjective function and the constraint sets are equivalent.


The deadline constraints in (5.4b) require that given any r and any k ≤ r, the total traveltime from σk to σr+1 should be no greater than T in

r+1 − T outk . Comparing the formulation

79

in (5.4) with that in (5.2), although the number of constraints of the former is O(K2) whilethat of the latter is O(K), the former formulation can help us develop an efficient dual-basedheuristic SPEED.

5.5.2 The Lagrangian Dual Relaxation of MEET

We relax the deadline constraints in (5.4b) to the objective function by introducing a La-grangian dual variable λrk for each (k, r) pair

L(~x,~t, ~λ

),

K∑r=1

r∑k=1

λrk ·

(r∑j=k

∑e∈E

xejtej − T in

r+1 + T outk

)+

K∑i=1

∑e∈E

xei · ce (tei , ρi) .

We observe that the dual variable λrk in the above Lagrangian function is only associated withthe travel time of tasks from τk to τr. Therefore, we have that λrk is associated with τi if andonly if k ≤ i ≤ r, and the set of the dual variables associated with τi is λrk : ∀k ≤ i, ∀r ≥ i.According to this critical observation, the following holds

K∑r=1

r∑k=1

λrk

r∑j=k

∑e∈E

xejtej =

K∑i=1

∑e∈E

xei tei

K∑r=i

i∑k=1

λrk, (5.5)

since both sides of the equation are the sum of the dual variable times the task fulfilling timeover all correlative tasks and dual variables. With above equation, our Lagrangian functioncan be presented below

L(~x,~t, ~λ

)=

K∑i=1

∑e∈E

xei ·

[ce(t

ei , ρi) + tei

K∑r=i

i∑k=1

λrk

]−

K∑r=1

r∑k=1

λrk(T inr+1 − T out

k

).

With the following definition of µi

µi ,K∑r=i

i∑k=1

λrk, (5.6)

our Lagrangian function is

L(~x,~t, ~λ

)=

K∑i=1

∑e∈E

xei · [ce(tei , ρi) + teiµi]−K∑r=1

r∑k=1


k

).

The dual problem of MEET is thus as follows

max~λ≥0

D(~λ)

: D(~λ), min

~x,~tL(~x,~t, ~λ

),

where D(~λ) is the dual function.

80

Theorem 23.

D(~λ)

= −K∑r=1

r∑k=1


k

)+

K∑i=1

∑e∈p(µi)

wei (µi),

wherewei (µi) , ce (tei (µi), ρi) + µit

ei (µi), (5.7)

withtei (µi) , arg min

tle≤tei≤te(ρi)(ce(t

ei , ρi) + µit

ei ) . (5.8)


In Thm. 23, tei (µi) is the travel time minimizing the penalized edge cost with a price µiimposed on the edge travel time given specific µi. w

ei (µi) is the optimal penalized edge cost

including the fuel consumption cost and the travel time cost given the price µi, and wedenote the minimal penalized cost path from σi to σi+1 as p(µi).

According to Thm. 23, given dual variables ~λ, we can figure out the value of D(~λ) by solvingK shortest path problems independently using standard techniques, e.g., the Dijkstra’s algo-rithm [23]. This observation introduced by Thm. 23 motivates us to construct near-optimalsolutions to MEET, by iteratively updating dual variables to minimize duality gap.

5.5.3 Our Proposed Heuristic SPEED

We further define δ(µi) as the path travel time of p(µi), i.e.,

δ(µi) ,∑

e∈p(µi)

tei (µi). (5.9)

Similar to [18, Thm.3], we have the following theorem.

Theorem 24. δ(µi) is non-increasing in µi ∈ [0,+∞) for any i = 1, 2, ..., K.

Proof. Similar to [18, Thm.3] and is skipped.

Now we give a set of complementary-slackness-like conditions. We can construct an optimalsolution to MEET if the conditions are met.

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Theorem 25. Suppose the λrk,∀r = 1, 2, ..., K,∀k = 1, 2, ..., r, the µi,∀i = 1, 2, .., Kcomputed in (5.6), and the δ(µi),∀i = 1, 2, ..., K) computed in (5.9) meet the followingconditions [

T outk − T in

r+1 +r∑j=k

δ(µj)

]+

λrk

= 0,∀r = 1, 2, ..., K, ∀k = 1, 2, ..., r, (5.10)

where the function [f ]+g is defined as

[f ]+g =

maxf, 0, if g ≤ 0;f, otherwise.

Then each p(µi) defines a path for fulfilling τi with a travel time of tei (µi) assigned to eachedge e ∈ p(µi), and this solution must be optimal to MEET.


Our SPEED (see Algorithm 6) uses a sub-gradient based heuristic scheme to iterativelyupdate dual variables towards meeting the conditions in (5.10):

λrk = φ(λrk) ·

[T outk − T in

r+1 +r∑j=k

δ(µj)

]+

λrk

,∀r = 1, 2, ..., K,∀k = 1, 2, ..., r, (5.11)

where φ(λrk) is a positive step size to update λrk, based on Thm. 24.

Theorem 26. For any theoretically-feasible MEET instance, Algorithm 6 must give a feasiblesolution p satisfying all the time window constraints. The gap between the cost of p, namelyc(p), and the optimal cost, namely OPT, must be bounded above as follows:

c(p)− OPT ≤

0,

K2 ·max∀r,k

∣∣∣ ˙λrk∣∣∣ ·max∀r,kλrk/φ(λrk)

,

where the zero gap is for p given in Line 13, and the problem-dependent gap is for p givenin Line 18. λrk,∀r = 1, ..., K,∀k = 1, ..., r is the set of dual variables which define thesolution p.


Based on Thm. 26, Algorithm 6 always obtains feasible solutions to MEET with theoreticalperformance guarantee. Moreover, the achieved solution must be optimal if it is returned inLine 13. The optimality gap of Thm. 26 also can help us obtain near-optimal solutions quickly

82

Algorithm 6 SPEED(G,~σ)

1: procedure2: ite=1, p = NULL, c(p) = +∞, λrk = λrk = λmax,∀r = 1, 2, ..., K,∀k = 1, 2, ..., r.

3: while ∃ r, k :∣∣∣λrk∣∣∣ > tol and ite ≤ ITE do

4: for i = 1, 2, ..., K do5: Set µi according to equality (5.6)6: Obtain tei (µi),∀e ∈ E according to equality (5.8)7: Set wei (µi),∀e ∈ E according to equality (5.7)8: Get the shortest path p(µi) with wei (µi) from σi to σi+1

9: ph = p(µi), i = 1, 2, ..., K, ite = ite + 110: Set λrk, ∀r,∀k according to equality (5.11)11: Set λrk = λrk + λrk,∀r = 1, 2, ..., K,∀k = 1, 2, ..., r12: if λrk = 0,∀r = 1, 2, ..., K,∀k = 1, 2, ..., r then13: return p = ph

14: if ph is feasible and c(ph) < c(p) then15: p = ph

16: if ph is not feasible then17: λrk = λmax,∀r = 1, 2, ..., K,∀k = 1, 2, ..., r

18: return p

without waiting for the convergence of dual variables, as we can terminate Algorithm 6 oncethe optimality gap is below a user-defined tolerance.

Overall in this section, we develop an efficient heuristic SPEED for MEET, by iterativelyupdating dual variables to minimize the duality gap. We derive sufficient conditions underwhich the solution of SPEED must be optimal, and characterize a performance gap comparingthe solution of SPEED with the optimal when our sufficient conditions are not satisfied.


We use real-world traces to evaluate SPEED, by a comparison with fastest-/shortest- pathbaselines and a conceivable approach directly extending the state-of-the-art single-task al-gorithm [19]. We implement our simulations in C++ and run them on a laptop with a4-core Core-i5 (2.50GHz) processor and 16GB memory, running 64-bit Ubuntu 16.04 LTS.We use the SNAP graph [52] to construct the US national highway system from the clinchedhighway mapping project7 consisting of 84504 nodes and 178238 directed edges. Road gradeis obtained from the node elevation service provided by the elevation point query service8

7Clinched highway mapping, http://cmap.m-plex.com/8Elevation point query service, http://nationalmap.gov/epqs/

83

from the US geological survey. The maximum road driving speed rue is set to be the histori-cal average speed by collecting real-time speed data from HERE map9 for 2 weeks, and theminimum road driving speed rle is manually set to be rle = min30, rue . We use ADVISORsimulator [61] to collect fuel consumption rate data with driving speed given various roadgrade and truck load. We consider a class-8 heavy truck of Kenworth T80010. We use thecurve fitting toolbox in MATLAB to learn the fuel consumption rate function fe(re, ρ) thatis modeled as a 3-order polynomial function with speed given road grade and truck load.

The highway network is preprocessed first: (i) the graph is cut to the “eastern” part; (ii)we merge non-intersection roads into a single road segment, if they have the same gradelevel; and (iii) the “eastern” part is divided into 22 regions (see Fig. 5.1(b)), where the nodenearest to each region’s center is used as the candidate for the task source node and the taskdestination node, same as the experimental setting in [18,19]. We set all the earliest leavingtime constraints to be 0, i.e., T out

i = 0, i = 1, 2, ..., K. As for SPEED (i.e., Algorithm 6),we fix φ(·) to be 0.1, and fix the algorithm tolerance level tol to be 0.01. For the sake ofconvenience, we denote the minimal time of independently fulfilling the single task τi as t∗i ,

t∗i , min~xi∈Xi

(xei · tle

), i = 1, 2, ..., K. (5.12)

5.6.1 Fuel-Consumption-Rate Function Model

Given a road grade and a truck load, we assume the truck fuel consumption rate (gph) off(r) with the speed (mph) of r to be the following function in our simulations:

f(r) = a · r3 + b · r2 + c · r + d,

where a, b, c, d are parameters learned by the curve fitting toolbox in MATLAB based onthe simulated fuel consumption rate data collected from the ADVISOR simulator.

To figure out the fuel consumption rate data f with a driving speed r, a road grade θ, and atruck load ρ, we specify a driving cycle test of 4 hours with a constant speed of r, a constantroad grade of θ, and a constant truck load of ρ. We enumerate θ from −10.0 to 10.0 witha step of 0.1, r from 10 to 70 with a step of 0.2, and we set ρ to be 0% (empty load), 50%(half load), or 100% (full load).

We present a part of the fuel consumption rate results with driving speed in Fig. 5.2. InFig. 5.2(a), we fix the truck to be full-load and assume it drives on a road with grade of −1,0, or 1. We observe that large grade and high speed cause large fuel consumption rate,and we empirically verify that function f(r) is strictly convex in reasonable speed regions.In Fig. 5.2(b), we assume the truck has a load of 0%, 50%, or 100%, driving on a road withgrade of 0. Similar to Fig. 5.2(a), Fig. 5.2(b) also verifies the convexity of f(r). Besides,we observe that a heavy truck load leads to a large truck fuel consumption.

9HERE, https://www.here.com/10Kenworth T800, https://www.kenworth.com/trucks/t800

84

(a) Results with different road grade. (b) Results with different truck load.

Figure 5.2: Simulated truck fuel consumption rate with driving speed.

Table 5.4: Compare SPEED with alternatives for the instance of (1, 9, 22, 40, 65). A lowerbound of the optimal fuel consumption is 478.73 according to Thm. 26.Algorithm First task from 1 to 9 Second task from 9 to 22 Total performance

Time Distance Fuel Time Distance Fuel Time Incre. Distance Incre. Fuel Incre.Fastest 19.54 1306 276.5 24.48 1613 337.8 44.02 - 2919 0.07 614.3 28.32Shortest 19.56 1306 276.18 24.58 1611 338.34 44.14 0.27 2917 - 614.52 28.36PASO 39.93 1307 202.92 25.06 1613 329.94 64.99 47.64 2920 0.1 532.86 11.31SPEED 29.03 1307 215.25 35.96 1616 264.52 64.99 47.64 2923 0.21 479.77 0.22

5.6.2 Comparing SPEED with Alternatives for Two Tasks

We compare our heuristic SPEED with two baselines and a conceivable alternative approachfor fulfilling two tasks:

1. A fastest-path-based baseline: road driving speed re of each e ∈ E is fixed as themaximum one, i.e., re = rue , and each task τi is fulfilled by the path with the minimaltravel time from σi to σi+1. Note that there is no task execution times optimizationinvolved in this baseline.

2. A shortest-path-based baseline: road driving speed re for each e ∈ E is fixed as themaximum one, i.e., re = rue , and each τi is fulfilled using the path with the minimaltravel distance from σi to σi+1. Note that there is no task execution times optimizationinvolved in this baseline.

3. A PASO-based approach with greedy task execution times allocation: we greedily al-locate deadlines as large as possible for individual tasks from the first task to the lasttask one by one. For the ith task τi, we run the heuristic from [19] to solve a problemPASO, which minimizes the fuel consumption of fulfilling τi subject to a deadline ofT ini+1 − maxT out

i , ai−1. Here ai−1 is defined in formula (5.2), which is the truck ar-rival time at σi and can be achieved as we have solved the PASO problems for tasks

85

Table 5.5: Compare SPEED with alternatives for the instance of (18, 11, 4, 22, 45, 75, 105),where a lower bound of the optimal fuel consumption is 664.77 according to Thm. 26.

Algorithm First task from 18 to 11, Second task from 11 to 4 Third task from 4 to 22 Total performanceTime Distance Fuel Time Distance Fuel Time Distance Fuel Time Distance Fuel Fuel-Incre.

Fastest 24.5 1616 341.38 20.02 1401 183.57 26.52 1744 366.21 71.04 4761 891.16 34.06Shortest 24.5 1616 341.39 20.02 1401 183.57 26.59 1743 366.25 71.11 4760 891.21 34.06PASO 44.97 1616 254.18 30.02 1401 112.15 29.79 1746 330.72 104.78 4763 697.05 4.86SPEED 35.36 1616 269.11 31.45 1401 107.71 38.19 1747 288.62 105 4764 665.44 0.1

τ1, τ2, ..., τi−1 before solving the PASO problem for the ith task τi.

We consider an instance of (σ1, σ2, σ3, Tin2 , T

in3 ) = (1, 9, 22, 40, 65), i.e., there are two tasks

with the first task from 1 to 9 and the second task from 9 to 22, assuming full load for bothtasks and T in

2 = 40, T in3 = 65. We remark that this instance is representative because (i)

both tasks are heavy-duty (full load) and long-haul (both tasks require the truck to travelacross the US); and (ii) the latest arrival time constraints are larger than the minimal taskexecution times, allowing a large design space for optimizing task execution times to reducetruck fuel consumption.

We give the simulation results in Tab. 5.4, where we also give the increment (%) of the traveltime, travel distance, and fuel consumption of the four solutions as compared to the respectiveoptimal. From the table we observe that both the fastest solution and the shortest solutionconsume ∼ 30% more fuel than SPEED. The PASO solution saves fuel for the individual taskτ1 in the cost of a larger travel time compared with SPEED. However, its solution is far fromoptimal, due to its greedy and non-optimal task execution times allocation (∼ 40 hours forτ1 and ∼ 25 hours for τ2). The solution of SPEED is close-to-optimal and saves 10% fuelcompared with the PASO solution, with a close-to-optimal execution time allocated for eachtask (∼ 29 hours for τ1 and ∼ 36 hours for τ2).

Tab. 5.4 also suggests that the travel distance of solutions of the four algorithms are similar,but the travel time and fuel consumption of these solutions are different. This highlights theimportance of exploring speed planning and task execution times optimization, in additionto path planning, in reducing the fuel consumption under the multi-task setting.

5.6.3 Comparing SPEED with Alternatives for Three Tasks

We then simulate a three-task instance (σ1, σ2, σ3, σ4, Tin2 , T

in3 , T

in4 ) of MEET. We assume full

load for τ1 and τ3, but an empty load for τ2, corresponding to a practical setting where thetruck needs to fulfill two freight transportation requests (τ1 from σ1 to σ2, and τ3 from σ3 toσ4), and have to travel with an empty load from the destination (σ2) of the first request tothe source (σ3) of the second request.

Our instance is (18, 11, 4, 22, 45, 75, 105) and the corresponding simulation results are intro-duced in Tab. 5.5. SPEED obtains a close-to-optimal solution due to a close-to-optimal task

86

(a) We set T in2 = t∗1 · (1 + x%) and T in

3 =(t∗1 + t∗2) · (1 + x%).

(b) We set T in2 = T in

3 = (t∗1 +t∗2) ·(1+x%).

Figure 5.3: Fuel saving achieved by SPEED with the time window constraints, as comparedto the fastest-path-based baseline. Here t∗i is defined in the equation in (5.12).

execution time allocation (35.36 hours for τ1, 31.45 hours for τ2, and 38.19 hours for τ3). Incontrast, the PASO-based alternative fails to minimize fuel consumption efficiently due tothe non-optimal task execution times allocation.

5.6.4 Impact of Time Windows on the Fuel Consumption

We consider MEET instances of two tasks each with a full load, and estimate how timewindow constraints affect fuel consumptions of different algorithms.

First, we consider a setting of T in2 = t∗1 · (1+x%) and T in

3 = (t∗1 + t∗2) · (1+x%). We enumeratex from 20 to 100 with a step of 20. Given a specific x, we run 1000 simulations where ineach simulation, we randomly select σ1, σ2 6= σ1, and σ3 6= σ2. We give the fuel consumptionreduction results comparing our SPEED to the fastest-path-based approach in Fig. 5.3(a).We observe that SPEED can save more fuel with larger time window constraints.

In our simulations, the shortest-path baseline performs almost same as the fastest-pathbaseline. We give the fuel consumption reduction comparing SPEED with the PASO-basedalternative in Fig. 5.4, according to which the fuel consumption of SPEED is just slightlybetter than that of the PASO-based alternative. This is because that under the setting ofT in

2 = t∗1 · (1 + x%) and T in3 = (t∗1 + t∗2) · (1 + x%), the greedy task execution times allocation

of the PASO-based approach is near-optimal.

Next we consider a different setting of T in2 = T in

3 = (t∗1 + t∗2) · (1 + x%). We present thefuel consumption reduction comparing SPEED with the fastest-path baseline in Fig. 5.3(b),which is similar to Fig. 5.3(a)

Recall that the PASO alternative obtains close-to-optimal solutions (see Fig. 5.4) in the first

87

Figure 5.4: Fuel saving of SPEED as compared to the PASO-based approach. We set T in2 =

t∗1 · (1 + x%) and T in3 = (t∗1 + t∗2) · (1 + x%), where t∗i is defined in the equation in (5.12).

Table 5.6: Fuel consumption reduction of SPEED as compared to the PASO-based alternative.We set T in

2 = T in3 = (t∗1 + t∗2) · (1 + x%), where t∗i is defined in the equation (5.12).

Time window constraints increment x% 20% 40% 60% 80% 100%Ratio of solvable instances of the PASO alternative 1% 13% 68% 92% 100%

Fuel saving of SPEED on average? 9.2% 8.6% 5.9% 2.8% 0.3%?: The fuel saving of SPEED as compared to the PASO alternative is calculated only based

on solvable instances of the alternative, instead of all the 1000 instances.

setting. In sharp contrast, in the second setting where T in2 = T in

3 = (t∗1 + t∗2) · (1 + x%),the PASO-based approach cannot even obtain feasible solutions for many instances (seeTab. 5.6), due to the non-optimal greedy task execution times allocation: (i) the allocatedtask execution time of τ1 is way too large in order to minimize the fuel consumption offulfilling τ1; (ii) then the remaining deadline for fulfilling τ2 is smaller than t∗2 in manyinstances, and thus there are no feasible solutions. Note that our SPEED always obtainsclose-to-optimal solutions meeting all the time window constraints, by jointly allocating taskexecution times for the two tasks towards the optimal.

5.6.5 Fuel Saving of SPEED with the Number of Tasks

Finally, we conduce simulations to evaluate the empirical performance of SPEED with differ-ent number of tasks. For each experimental case characterized by a specific K ∈ 2, 3, ..., 6and an x ∈ 25, 50, we run 1000 simulations each with K full-load tasks defined by ran-domly selected source/destination nodes σ1, ..., σK+1. Latest arrival time constraints aresame for any σi, i = 2, ..., K + 1 : T in

i = t∗ · (1 + x%), where t∗ =∑K

i=1 t∗i is the minimal time

to fulfill those K tasks. We given the fuel reduction results of SPEED as compared to thefastest-path baseline in Fig. 5.5. The figure suggests that the fuel saving of SPEED is robustto the number of tasks (∼ 16% for x = 25 and ∼ 22% for x = 50).

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(a) Simulations for x = 25. (b) Simulations for x = 50.

Figure 5.5: The fuel reduction of SPEED as compared to the fastest-path baseline, as afunction of K. All latest arrival time constraints are set to be (1 + x%) ·

∑Ki=1 t

∗i .

Table 5.7: Ratio of solvable MEET instances of the PASO-based alternative with the numberof tasks. All latest arrival time constraints are set to be (1 + x%) ·

∑Ki=1 t

∗i .

K = 2 K = 3 K = 4 K = 5 K = 6x = 25 1% 0% 0% 0% 0%x = 50 27% 2% 0% 0% 0%

We give the number of solvable instances of the PASO alternative in Tab. 5.7. In sharpcontrast, we remark again that for all simulated instances, SPEED obtains near-optimalsolutions meeting time window constraints.

5.7 Chapter Summary

In this chapter we consider a scenario where in a national highway network a heavy truck istraveling to fulfill many ordered transportation tasks. We formulate a problem MEET, withan objective of minimizing fuel consumption under task pickup and delivery time windowconstraints. Optimizing task execution times is a new challenging design space for savingfuel in our multi-task setting, and it differentiates our study from existing ones that areunder the single-task setting. First we show that it is NP-hard to solve MEET optimally,and it is non-convex even to optimize task execution times by itself. Next we develop anefficient heuristic SPEED. We characterize sufficient conditions under which SPEED gives anoptimal solution, and derive an optimality gap of SPEED when the conditions are not met.

We simulate a truck driving across the US national highway system to empirically evalu-ate SPEED. As compared to the fuel consumption achieved by the fastest-/shortest- pathbaselines (resp. a conceivable approach generalized from the state-of-the-art single-task al-

89

gorithm), SPEED can reduce the fuel consumption by 22% (resp. 10%). Moreover, for allsimulated instances, SPEED obtains near-optimal solutions meeting time window constraints;as a comparison, the conceivable approach violates time window constraints for up to 45%of the instances in our simulations. We also observe that the fuel saving of SPEED is robustto the number of tasks.

Chapter 6

Dissertation Summary and FutureWork

6.1 Dissertation Summary

Nowadays a huge amount of critical routing applications, e.g., cloud video conferencing,mobile video recognition, and long-haul heavy-duty truck transportation, requires serviceswith low delay guarantee. Thus multi-path routing problems which minimize delays, orat least upper bound delays, of all traffic are fundamentally important both in the com-munication field and in the transportation field. We study four delay-aware multi-pathrouting problems, which either optimize an delay-aware objective or consider delay-basedconstraints. We design approximation algorithms with theoretical performance guarantee,to obtain high-quality multi-path routing solutions with low delay performance.

Minimize maximum delay and average delay. In the first problem, we consider a single-unicast scenario where in a multi-hop network a sender requires to use multiple paths tostream a flow at a fixed rate to a receiver. We focus on understanding the fundamental com-patibility of two sender-to-receiver delay metrics, i.e., the average delay and the maximumdelay. Existing results pessimistically suggest that a flow cannot minimize the two delaymetrics simultaneously within bounded-ratio gaps to the optimal. In comparison, we provethat there exist multiple flow solutions each of which can send (1 − ε)-portion of the ratewith the two delay metrics both within a (1/ε)-ratio gap to the optimal, for any ε ∈ (0, 1).The gap (1/ε) is proven to be at least near-tight, and we further show that our solutions canbe generalized to the multiple-unicast scenario.

Minimize Age-of-Information. In the second problem, we consider a single-unicast scenariowhere in a multi-hop network a sender requires to use multiple paths to send a batch ofdata to a receiver periodically. We study an NP-hard problem of minimizing AoI subjectto throughput requirements. Our AoI problem differs from existing ones in that we are

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the first to consider a multi-path communication setting and the batch generation of data.We design both an optimal algorithm with a pseudo-polynomial time complexity and anapproximation framework with a polynomial time complexity. Our framework can buildupon any polynomial-time α-approximation algorithm of the maximum delay minimizationproblem, e.g., the one from [64] with α = 1 + ε given any user-defined ε > 0, to constructan (α + c)-approximate solution for minimizing AoI. Here c is a constant dependent onthroughput requirements.

Maximize network utility. In the third problem, we consider a problem of maximizing net-work utilities subject to user throughput requirements and maximum delay constraints. Weassume that a user’s utility is a concave function of the experienced maximum delay or theachieved throughput, and each user can use multiple paths to send the data from its senderto its receiver over a multi-hop network. First, we observe that it is impossible either to gen-erate an arbitrary solution with all constraints satisfied, or to construct an optimal solutionafter we relax maximum delay constraints or throughput requirements, in a polynomial timeunless P = NP. We then develop a polynomial-time algorithm that obtains solutions withconstant approximation ratios under realistic conditions, at the cost of violating constraintsby up to constant-ratios.

Minimize fuel consumption for a heavy truck to timely fulfill transportation tasks. In thefourth problem, we consider a common truck operation scenario where a long-haul truck isdriving in a highway network to fulfill many ordered transportation tasks. We study a timelyeco-routing problem of minimizing fuel consumption under task pickup and delivery timewindow constraints. It requires us to jointly optimize path planning, speed planning, andtask execution times to obtain high-quality solutions. We note that task execution timesoptimization is a new challenging design space for saving fuel in our multi-task setting, andit differentiates our study from existing ones under the single-task setting. We prove that itis NP-hard to solve the problem optimally, and develop an efficient heuristic. Our heuristiccan generate an optimal solution under derived conditions, and we further characterize anoptimality gap for it when the conditions are not met. We simulate a truck driving acrossthe US national highway system to empirically evaluate our solutions. As compared to thefastest-/shortest- path baselines, our solutions can reduce the fuel consumption by 22%,which is independent to the number of tasks.

6.2 Future Work

Delay-aware multi-path routing is an important yet less explored research direction. In thisdissertation we study four fundamental problems in this area, and leave many problems,some of which are natural generalizations of our studied problems, as the future work. Thefollowing introduces a small set of future research directions that arise from our study.

Minimize maximum delay and average delay. In Chapter 2, we design multi-path routing

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algorithms to minimize maximum delay and average delay both within constant-ratio gapsto the optimal. As suggested by our simulation results over real-world traces, empirical delaygaps are much smaller than our proposed theoretical constant-ratio results. It is a futuredirection of characterizing better (tighter) gaps on the two delay metrics, with more networkparameters (the number of nodes/edges, link delay models, etc.) taken into account. Withtight delay gap results, one can learn the best maximum delay and the best average delaythat can be achieved once given a network.

Minimize Age-of-Information. In Chapter 3, we develop multi-path routing algorithms tominimize AoI with throughput requirements, for supporting one periodic transmission task.Considering that the problem is under a single-unicast communication setting, it is a naturalfuture direction of studying similar AoI minimization problems under a multiple-unicast set-ting, i.e., for supporting multiple periodic transmission tasks with different sender-receiverpairs. It is obvious that multiple-unicast AoI minimization fits real-world applications bet-ter. However, it is non-trivial to extend our study from the single-unicast setting to themultiple-unicast setting, as in the multiple-unicast scenario different tasks can have differenttask activation periods, introducing a new design space of coordinating activation periodsoptimally among tasks to minimize AoI.

Maximize network utility. In Chapter 4, we present multi-path routing algorithms to maxi-mize aggregate throughput-/delay- based user utilities subject to maximum delay constraintsand user throughput requirements. Our algorithms provide constant approximation ratios,at a cost of violating delay constraints or throughput requirements by constant ratios. OurNUM study assume a constant link delay model which ignores many practical link trans-mission concerns. It is a promising future work of maximizing network utility under morepractical link delay models, e.g., the traffic-dependent link delay model, or the wireless linkdelay model that considers wireless interference constraints and packet loss ratio.

Minimize fuel consumption for a heavy truck to timely fulfill transportation tasks. In Chap-ter 5, we propose an efficient heuristic for minimizing truck fuel consumption for fulfillingmultiple transportation tasks under task pickup and delivery time window constraints. Notethat the fulfilling order of transportation tasks are given in our study. It is a possible futurework of optimizing task fulfilling orders to further save fuel. Besides in our truck transporta-tion study, speed planning which is one of our design spaces is subject to static road drivingspeed limits. However, real-world transportation traffic is dynamic and time-dependent.Hence it is another possible future work of studying similar energy-efficient timely trucktransportation problems under dynamic road driving speed limits. Both future directionscan be of particular interest for truck operators.

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Appendices

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Appendix A

Proofs of our Theorems and Lemmas

A.1 Proof of our Lem. 1

Proof. Suppose Algorithm 1 gives a path-defined system-optimal flow fSO(R) in Line 4.According to Algorithm 1, after we iteratively delete (ε ·R) rate from fSO(R), we obtain thesolution fSF[(1 − ε)R] in the end. Let us denote fl as the flow at the beginning of the l-thiteration (at the end of the (l − 1)-th iteration), and we assume 1 ≤ l ≤ L + 1. Obviously,f1 = fSO(R), fL+1 = fSF[(1− ε)R].

Suppose Pl is the set of of flow-carrying paths in fl, and pl ∈ Pl is the slowest flow-carryingpath in Pl. Algorithm 1 requires that we delete a rate of xl > 0 from pl, in the l-th iteration.

Since delay functions are non-decreasing, the path delay of any flow-carrying path will notincrease when deleting flow rate. Thus,

M(fl+1) ≤ M(fl), (A.1)

implying thatM(fSF[(1− ε)R]) ≤M(fSO(R)). (A.2)

We have the following in terms of the total delay, for any l

T (fl) =∑e/∈pl

[xeDe(xe)] +∑e∈pl

[xeDe(xe)] (A.3a)

=∑e/∈pl


[(xe − xl)De(xe)] + xl∑e∈pl

De(xe) (A.3b)

(a)=∑e/∈pl


[(xe − xl)De(xe)] + xlM(fl) (A.3c)

(b)

≥ (∑e/∈pl


[(xe − xl)De(xe − xl)]) + xlM(fl) (A.3d)

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103

Figure A.1: A two-node two-link network. We assume that D(x) = xp (p > 1) is the delayfunction of the solid link, and D(x) = 1 is the delay function of the dashed link.

(c)= T (fl+1) + xlM(fl)

(d)

≥ T (fl+1) + xlM[fSF((1− ε)R)]. (A.3e)

Definition of pl implies the equality in (a). The inequality in (b) holds because link delayfunctions are non-decreasing. The equality in (c) holds due to the definition of fl+1. Theinequality in (d) comes from (A.1) and fL+1 = fSF[(1− ε)R]. We then do summation for theinequality in (A.3) over l ∈ [1, L], and get

T ∗(R) = T [fSO(R)] = T (f1) ≥ T (fL+1) +

(L∑l=1

xl

)· M[fSF((1− ε)R)]

= T [fSF((1− ε)R)] + ε ·R · M[fSF((1− ε)R)],

which implies that M[fSF((1− ε)R)] ≤ A∗(R)/ε holds.

For the average delay, we have

A(fl) =T (fl)

|fl|≥ T (fl+1) + xlM(fl)

|fl|//by (A.3)

=A(fl+1)(|fl| − xl) + xlM(fl)

|fl|

=A(fl+1)|fl|+ xl(M(fl)−A(fl+1))

|fl|

≥ A(fl+1)|fl|+ xl(M(fl+1)−A(fl+1))

|fl|//by (A.1)

≥ A(fl+1)|fl||fl|

= A(fl+1),

which proves that A[fSF(1− ε)R] ≤ A∗(R), by iterating l from 1 to L.

A.2 Proof of our Thm. 5

Proof. Given any ε ∈ (0, 1), let β = ε1−ε , or ε = β

1+βequivalently. Due to [74, Thm. 3.2], we

have

T (fNE[(1− ε)R]) ≤ 1

βT (fSO[(1 + β)(1− ε)R]) =

1− εεT (fSO(R)). (A.4)

104

With (A.4), the following holds for fNE[(1− ε)R]

A(fNE[(1− ε)R]) =T (fNE[(1− ε)R])

(1− ε)R≤

1−εεT (fSO(R))

(1− ε)R=

1

εA(fSO(R)) =

1

εA∗(R),

which proves the average delay gap in our Thm. 5.

The tightness proof of the gap is also motivated by [74, Thm. 3.2]. We consider an instancegiven by Fig. A.1. We set the flow rate requirement to be R = 1

1−ε . In this instancefNE[(1− ε)R] will allocate all rate 1 to the lower link, thus A(fNE[(1− ε)R]) = 1.

We can figure out fSO(R) by solving the following convex program

minx∈[0,R]

(R− x) + xp+1,

where x is the flow rate assigned to the lower solid link. We observe that the optimal solution

is x∗ = (p+ 1)−1p ∈ (0, 1) ⊂ [0, R]. Thus we have

A∗(R) =(R− x∗) + (x∗)p+1

R=R− (p+ 1)−

1p + (p+ 1)−

p+1p

R.

Since limp→+∞

(p+ 1)−1p = 1 and lim

p→+∞(p+ 1)−

p+1p = 0, we have

limp→+∞

A∗(R) =R− 1

R=

1/(1− ε)− 1

1/(1− ε)= ε.

Therefore,

limp→+∞

A(fNE[(1− ε)R])

A∗(R)=

1

ε.

A.3 Approximate Min-Max Flow in Simulations

We find a min-max flow fMM(R) in a sub-network of GEANT. It is clear that we haveM(fMM(R)) ≥ M(fMM(R)). To estimate the quality of using fMM(R) to approximatefMM(R), we further construct a lower bound of M(fMM(R)).

A.3.1 Figure Out an Upper Bound of M(fMM(R))

We construct a sub-network of GEANT with all the nodes but only the following links:

(IS, UK), (IS,DK), (UK, IL), (UK,CY ), (CY,DE), (DE, IL), (DK,DE),

105

UK

IS DK

DE

IL

Source: Iceland

Receiver: Israel

Figure A.2: GEANT∗: a network reduced from GEANT.

(DK,NL), (NL,BE), (BE,UK), (UK,FR), (FR,CH), (CH,DE).

Since the sub-network has a small size, we can find the min-max flow its by enumerating pos-sible. We denote the min-max flow in this sub-network as fMM(R). Obviously, M(fMM(R))is an upper bound of M(fMM(R)).

A.3.2 Figure Out a Lower Bound of M(fMM(R))

We observe that the network GEANT can be reduced to a much simpler one shown inFig. A.2 that is denoted as GEANT∗. In the figure, solid links are the links belonging toGEANT, while dashed links represent that multiple paths from DK to UK (resp. from UKto DE, and from DK to DE) exist in GEANT.

For any flow f using simple paths to send traffic from IS to IL in GEANT, it correspondsto a feasible flow g from IS to IL in GEANT∗: flow rate in g assigned on (DK,UK) (resp.(DK,DE) and (UK,DE)) corresponds to the flow rate in f from (IS,DK) (resp. (IS,DK)and (IS, UK)) that will finally use (UK, IL) (resp. (DE, IL) and (DE, IL)) to go to receiverIL. Now we figure out capacities for dashed links such that the min-max flow from IS to ILin GEANT∗, denoted as fMM(R), can provide a maximum delay which is a lower bound forthe minimal maximum delay in GEANT.

For the node UK in GEANT, any flow rate x from UK to DE must use one or more linksin the set of (UK,PT ), (UK,FR), (UK,BE), (UK,CY ). Since clearly that the minimalmaximum delay for a rate x to travel using four parallel links each with a capacity of 100must be 1/(100 − x/4), the delay of the flow rate x assigned to the dashed link (UK,DE)in GEANT∗ must be lower bounded by 1/(100− x/4). Considering that

1

100− x4

=4

400− x= 4 · 1

400− x,

In GEANT∗ for the dashed link from UK to DE, we represent it by four serial links, eachwith a capacity of 400. Similarly, the dashed link from DK to UK is also represented by four

106

serial links, each with a capacity of 400, and same to the dashed link from DK to DE. Bysuch defined capacities of dashed links of GEANT∗, considering flow f in GEANT and itsassociated flow g in GEANT∗, clearly that M(g) ≤ M(f) where M(f) is the maximumdelay of f in GEANT, and M(g) is the maximum delay of g in GEANT∗.

Now it holds that M(fMM(R)) ≤M(fMM(R)), due to the following inequalities:

M(fMM(R)) ≤M(gMM(R)) ≤M(fMM(R)),

where the first inequality holds due to that fMM(R) minimizes the maximum delay inGEANT∗, and the second inequality holds due to that M(g) ≤M(f).

A.3.3 Compare the Upper Bound with the Lower Bound

Now we carry out extensive simulations to obtain the maximum delay of fMM(R) in GEANTand the maximum delay of fMM(R) in GEANT∗ for all possible R from 0 to 10 with a step of0.1. The average (resp. maximum, minimum) relative error on the maximum delay is 1.73%(resp. 4.58%, 0.21%), where the relative error is defined as:

Relative Error =M(fMM(R))−M(fMM(R))

M(fMM(R)).

Therefore, fMM(R) is a good approximation to fMM(R).

A.4 Proof of Our Thm 14

Proof. We assume that fSO(R) is the path-defined system optimal flow, and fSF-2[(1− ε)R]is achieved by deleting (ε ·Rk) rate from the slowest flow-carrying paths of unicast k of thisfSO(R), for each k = 1, 2, ..., K.

For a multiple-unicast flow f = fk, k = 1, 2, ..., K, because a link can be used by differentunicasts, it is obvious that deleting flow rate of f i not only can impact the delay of the unicasti, but also may impact the delay of another unicast j where j 6= i. However, according tothe definitions of M(f j), T (f j), and A(f j), since link delay functions are non-decreasing,it is clear that none of M(f j), T (f j), or A(f j) can increase when we delete flow rate of f i.Based on this observation, the following holds due to a similar proof of our Lem. 1

A(fkSF-2[(1− ε)R]

)≤ A

(fkSO(R)

), ∀k = 1, ..., K, (A.5)

M(fkSF-2[(1− ε)R]

)≤ A

(fkSO(R)

)/ε,∀k = 1, ..., K. (A.6)

107

Then we can prove the average delay gap as follows

A(fSF-2[(1− ε)R])(a)

≤ A(f iSF-2[(1− ε)R]

) (b)

≤ A(f iSO(R)

)≤ (R/Rmin) · A

(fSO(R)

)= (R/Rmin) · A∗ (R) ,

where in (a) we assume A(f iSF-2[(1 − ε)R]) = max1≤k≤K A(fkSF-2[(1 − ε)R]), and (b) comesfrom (A.5).

Similarly, we can prove the maximum delay gap using the inequality (A.6)

M(fSF-2[(1− ε)R]) =M(f iSF-2[(1− ε)R]

)≤ A

(f iSO(R)

)/ε ≤ (R/Rmin) · A∗(R)/ε

≤ (R/Rmin) · A(fMM(R))/ε ≤ (1/ε) · (R/Rmin) · M∗(R).

A.5 Slotted Transmission Model Considers Different

Kinds of Networking Delays

Similar to the discussions in [6], we remark that different kinds of networking delays can betaken into account by our slotted data transmission model.

(1) Link propagation delay. It is the ratio comparing the link length with the signal propa-gation speed. Clearly that our link delay de can consider such a non-negative constant.

(2) Link transmission delay. In general, it is given by Se/be where Se is the data size. Inour slotted data transmission model, if Se ≤ be, the link transmission delay is dSe/bee = 1and is counted into de. Otherwise if be < Se ≤ 2be, the bandwidth constraint requires that(i) at time t, e allocates be data to it, experiencing a transmission delay of 1; (ii) then attime t+ 1, e allocates the remaining Se − be data to it, experiencing a transmission delay of2, where 1 of the 2 delay is the data-holding delay from time t to time t + 1. For arbitrarySe : (k − 1)be < Se ≤ kbe,∀k ∈ Z+, we can calculate associated transmission delay followinga similar procedure.

(3) Node queuing delay. Before being transmitted, data has to be processed first whenarriving at a router (node). Since the service rate λv of a router is finite, data has to bestored in the queue if the arrival rate is greater than λv. Our delay model can considerqueuing delay, which is the time of storing data in a router, due to the following concern:we can add a new node v′ to V , add a new link (v, v′) with a bandwidth of λv and a delay of1 to E, and change all the outgoing links of v to be the outgoing links of v′, for each routernode v ∈ V . Following the aforementioned discussions in (2), we observe that the updatednetwork can consider the queuing delay at the router v.

108


Proof. Given a path p and an offset ~u, let us denote xp(~u)(g) as the value of xp(~u) in g, .Suppose there exists a xp(~u) > 0 in g with k ·T (g) ≤ ui−dei−1

− ui−1 < (k+1) ·T (g), k ∈ Z+

for certain p = p, ~u = ~u, and i ∈ 1, 2, ..., |p|. In this proof we construct another solution fbased on g directly, with all results introduced by this lemma satisfied.

Set xp(~u)(f) = 0 for all ~u and all p, and let T (f) = T (g). Then for each positive xp(~u)(g),(i) if 〈p, ~u〉 6= 〈p, ~u〉, we let xp(~u)(f) = xp(~u)(f) + xp(~u)(g). (ii) If 〈p, ~u〉 = 〈p, ~u〉, welet xp(~u∗)(f) = xp(~u∗)(f) + xp(~u)(g), where comparing ~u∗ = u∗j , j = 0, 1, ..., |p| with~u = uj, j = 0, 1, ..., |p|, we have u∗j = uj for j = 0, 1, ..., i − 1, but u∗j = uj − k · T (g) forj = i, i+ 1, ..., |p|.

Considering that T (f) = T (g) and g meets throughput requirements, we have that f meetsthroughput requirements. Considering that the difference between u∗j and uj is a multipleof the task activation period T (g), we note that f meets link bandwidth constraints, sinceif xp(~u)(g) respects the bandwidth constraint of e at certain offset j ∈ 0, 1, ..., T (g) − 1,xp(~u∗)(f) must respect the bandwidth constraint of e at the same offset j, for any link e ∈ p.

Next, consider that we have assumed that xp(~u)(g) > 0 but with k ·T (g) ≤ ui−dei−1−ui−1 <

(k + 1) · T (g), k ∈ Z+ for a specific i ∈ 1, 2, ..., |p|. By the definition of f , the followingholds: (i) for any 〈p, ~u〉 6= 〈p, ~u∗〉 such that xp(~u)(g) = 0, we have xp(~u)(f) = 0; (ii)it holds that xp(~u)(g) > 0, while xp(~u)(f) = 0; (iii) although xp(~u∗)(f) > 0, we haveu∗i − dei−1

− u∗i−1 ≤ T (g)− 1, and u∗j ≤ uj for any j = 0, 1, ..., |p|. Those three results imply(i) based on Lem. 4, we have M(f) ≤ M(g), Λp(f) ≤ Λp(g), and Λa(f) ≤ Λa(g); (ii) forcertain node of g, the data-holding at certain offset is greater than T (g) − 1, while for thesame node at the same offset, it is reduced to be upper bounded by T (f)− 1 in f .


Proof. Following an arbitrary periodically repeated solution f , by its definition, we knowthat the batch of data generated at the sender at time k · T (f) will arrive at the receiver attime k·T (f)+M(f); similarly, the batch of data generated at the sender at time (k+1)·T (f)will arrive at the receiver at time (k + 1) · T (f) +M(f). For AoI, by its definition, it isM(f) both at time k · T (f) +M(f) and at time (k+ 1) · T (f) +M(f), while from the timek · T (f) +M(f) to the time (k + 1) · T (f) +M(f)− 1 it increases linearly from M(f) toM(f) + T (f)− 1. f thus has a peak AoI of M(f) + T (f)− 1, and an average AoI of

Λa(f) =

∑t∈Z I(f, t)∑

t∈Z 1= lim

k→+∞

k ·∑T (f)−1

i=0 (M(f) + i)

k · T (f)=M(f) +

T (f)− 1

2.

109

s r

Figure A.3: A two-node two-link network. We assume that Ru = 10/7, Rl = 1, D = 10, thebandwidth of e1 (resp. e2) is 1 (resp. 10), and the delay of e1 (resp. e2) is 1 (resp. 11).


Proof. We first prove the result 1 by contradiction.

Suppose R∗p < R∗m for certain R∗p ∈ ~Rp and R∗m ∈ ~Rm. Because R∗m minimize maximumdelay, the following holds

MR∗m ≤ MR∗p . (A.7)

Because R∗p minimizes peak AoI, according to Lem. 4, the following holds

MR∗p +D/R∗p − 1 ≤ MR∗m +D/R∗m − 1,

further implying that

MR∗p −MR∗m ≤ D/R∗m −D/R∗p(a)< 0, (A.8)

where the inequality (a) holds due to R∗p < R∗m. Since the inequality in (A.7) contradictswith that in (A.8), the following holds

minRp∈~Rp

Rp ≥ maxRm∈~Rm

Rm.

Following the same method, we have

minRa∈~Ra

Ra ≥ maxRm∈~Rm

Rm.

Now consider one example in Fig. A.3. Given T = 10, we have M1 = 10, Λ1p = 19, and

Λ1a = 14.5, since we can allocate 1 data to e1 at offsets i : i = 0, 1, ..., 9. Given T = 9, we

have M10/9 = 11, Λ10/9p = 19, and Λ

10/9a = 15, since we can allocate 1 data to e1 at offsets

i : i = 0, 1, ..., 8, and another 1 data to e2 at the offset 0. Similarly, we have M10/8 = 11,Λ

10/8p = 18, and Λ

10/8a = 14.5. And we haveM10/7 = 11, Λ

10/7p = 17, and Λ

10/7a = 14. Overall,

we have ~Rp = ~Ra = 10/7, different from ~Rm = 1.

110

s re2

e1

e3

Figure A.4: A two-node three-link network. The bandwidth of each link is 1, the delay ofe1, e2, and e3 are 1, 6, and 7, respectively, D = 5, Rl = 1, and Ru = 5/2.


Proof. First, for any Rm ∈ ~Rm and Rp ∈ ~Rp, the following holds

ΛRmp − ΛRp

p =MRm +D/Rm − 1−(MRp +D/Rp − 1

)=(MRm −MRp

)+D/Rm −D/Rp

(a)

≤ D/Rm −D/Rp ≤ D/Rl −D/Ru,

which proves the gap in (3.9). Considering that Rm minimizes maximum delay, the inequalityin (a) holds. We can prove the gap in (3.10) in the same way.

Second, for the gap (3.9), if D/Rl − D/Ru ≤ 1, it is clear that it is near-tight. In thefollowing we focus on instances assuming D/Rl −D/Ru ≥ 2.

Given any D > 0, D/Rl ∈ Z+, and D/Ru ∈ Z+, we consider a network with the sametopology as Fig. A.3, but assume the maximum throughput requirement is Ru/Rl, the min-imum throughput requirement is 1, the size of the data batch is D/Rl, the bandwidth of e1

(resp. e2) is 1 (resp. D/Rl), and the delay of e1 (resp. e2) is 1 (resp. D/Rl + 1). It canbe verified that in this example, we have M1 = D/Rl and Λ1

p = 2D/Rl − 1; and we have

MD/(Rl·T ) = D/Rl + 1 and ΛD/(Rl·T )p = D/Rl + T , for any T ∈ [D/Ru, D/Rl − 1], T ∈ Z+.

We observe that ~Rm = 1 and ~Rp = Ru/Rl.

ΛRmp − ΛRp

p = 2 · DRl

− 1−(D

Rl

+D

Ru

)≥ D

Rl

− D

Ru

− 1.

The near-tightness of the gap (3.10) can be proved using the same instance.


Proof. First we prove Rp ≥ Ra and MRp ≥ MRa by contradiction. Suppose R∗p < R∗a for

certain R∗p ∈ ~Rp and R∗a ∈ ~Ra. Based on Lem. 4, the following holds due to that R∗p minimizes

111

peak AoI, ,MR∗p +D/R∗p − 1 ≤ MR∗a +D/R∗a − 1. (A.9)

Similarly, because R∗a minimizes average AoI, the following holds

MR∗p +D/R∗p − 1

2≥ MR∗a +

D/R∗a − 1

2. (A.10)

The above inequality (A.9) implies that

D/R∗a −D/R∗p ≥ MR∗p −MR∗a . (A.11)

Now consider the following inequality

MR∗p +D/R∗p − 1

2−(MR∗a +

D/R∗a − 1

2

)=(MR∗p −MR∗a

)−D/R∗a −D/R∗p

2

(a)

≤D/R∗a −D/R∗p

2

(b)< 0, (A.12)

where inequality (a) comes from inequality (A.11), and the inequality (b) holds due toR∗p < R∗a. We observe that the inequality in (A.10) is contradicted with that in (A.12),

implying that we have Rp ≥ Ra for all Rp ∈ ~Rp and Ra ∈ ~Ra.

For maximum delay, for any Rp ∈ ~Rp and Ra ∈ ~Ra, the following holds

MRa −MRp = ΛRaa − (D/Ra − 1)/2−

(ΛRpa − (D/Rp − 1)/2

)=(ΛRaa − ΛRp

a

)+ (D/Rp −D/Ra)/2

(a)

≤ 0,

where (a) holds because (i) Ra minimizes average AoI, and (ii) Ra ≤ Rp holds as provedpreviously.

Second, we prove the gap (3.11) and (3.12). Because any Rp ∈ ~Rp minimizes peak AoI, thefollowing holds

MRp +D/Rp − 1 ≤MRa +D/Ra − 1,∀Rp ∈ ~Rp,∀Ra ∈ ~Ra,

implying thatD/Ra −D/Rp ≥MRp −MRa ,∀Rp ∈ ~Rp,∀Ra ∈ ~Ra. (A.13)

Therefore, for any Rp ∈ ~Rp and Ra ∈ ~Ra, we have

ΛRpa − ΛRa

a =MRp +D/Rp − 1

2−(MRa +

D/Ra − 1

2

)=(MRp −MRa

)+D/Rp −D/Ra

2

112

(a)

≤ D/Ra −D/Rp

2≤ D

2Rl

− D

2Ru

,

where inequality (a) holds due to inequality (A.13). The gap (3.12) holds due to similar

reasons, together with an observation that ΛRap − Λ

Rpp must be an integer due to Lem. 4 as

well as our slotted data transmission model.

Third, we prove the existence of an instance where Rp 6= Ra for any Rp ∈ ~Rp and Ra ∈ ~Ra,using the example in Fig. A.4.

We can verify that in Fig. A.4, we have M1 = 5, Λ1p = 9, and Λ1

a = 7; M5/4 = 6, Λ5/4p = 9,

and Λ5/4a = 7.5; M5/3 = 7, Λ

5/3p = 9, Λ

5/3a = 8; M5/2 = 7, Λ

5/2p = 8, and Λ

5/2a = 7.5. We

observe that in this example ~Rp = 5/2 and ~Ra = 1, which are different.


Proof. Only if part. Let us assume that there is a feasible periodically repeated solutionf = xp(~u) : ∀p,∀~u, where M(f) ≤ M and T (f) = D/R. Now we construct a flow f inGexp which is feasible to the linear program (3.13) from f , with the value of f lower boundedby D. Let the sender (resp. receiver) of f is s0 (resp. rM). Based on each positive xp(~u)of f where we assume p = 〈v1, v2, ..., v|p|〉 and ~u = u0, u1, ..., u|p|, (i) if the following linksbelong to Eexp, we allocate a rate of xp(~u) to each of them(

vuii , vui+deii+1

), i = 1, 2, ..., |p| − 1.

(ii) If the following links belongs to Eexp, we allocate a rate of xp(~u) to each of them(vui+dei+j

i+1 , vui+dei+j+1

i+1

), j = 0, ..., ui+1 − ui − dei − 1, i = 0, ..., |p| − 2.

(iii) Finally, we allocate a rate of xp(~u) to each of the following links(vui+dei+j

i+1 , vui+dei+j+1

i+1

), j = 0, ...,M − ui − dei − 1, i = |p| − 1.

It is straightforward that f satisfies the flow conservation constraints from s0 to rM in Gexp.The value of f is lower bounded by D as f satisfies the throughput requirements (3.7). Itholds that xe(i) ≤ be, ∀e ∈ E,∀i = 0, 1, ..., D/R− 1 in f , since f satisfies the link bandwidthconstraints (3.8). Therefore, the aggregate traffic rate of e(i) is upper bounded by be, for alle ∈ E and i = 0, 1, ..., D/R− 1. Overall, f is feasible to the linear program (3.13), with thevalue of f lower bounded by D, proving the only if part.

If part. It follows a similar proof. Let us assume that the solution given by linear pro-gram (3.13) has a value lower bounded by D. Although this solution may be defined on

113

s r

Figure A.5: A two-node two-link network. We assume that the delay of e1 (resp. e2) is 1(resp. d), the bandwidth of e1 (resp. (e2)) is 1 (resp. 5), D = 5, Rl = 5/6, and Ru = 5/3.

edges, we can figure out an equivalent path-defined solution f , i.e., f = xp,∀p ∈ Pexpwhere xp is the rate assigned to p, and Pexp is the set of simple paths from s0 to rM , af-ter flow decomposition. Note that |Eexp| upper bounds the number of flow-carrying pathsp ∈ Pexp [24]. We now construct a feasible periodically repeated solution f withM(f) ≤Mand T (f) = D/R, directly from the solution f .

Note that in f , both nodes vni and vmi with n < m can belong to a same path p. However, inthis case, we can figure out another flow equivalent to f , by allocating the rate xp originallyallocated to one outgoing link of vni now to (vn+j

i , vn+j+1i ), j = 0, 1, ...,m−n−1, implying that

for any xp > 0 in f , we can always figure out a path pf ∈ P corresponding to a path p ∈ Pexp.Now let xpf (~u) = xp where u0 = 0, u|pf | = u|pf |−1 + de|pf |−1

, and ui ∈ ~u, i = 1, 2, ..., |pf | − 1

to be the u ∈ [0, D], u ∈ Z such that this path p does not use (vui , vu+1i ) but uses certain

incoming link of vui . We can prove that this f meets bandwidth constraints and throughputrequirements, withM(f) ≤M and T (f) = D/R. Note that the size of f is no greater than|Eexp|, since the size of f is no greater than |Eexp|.


Proof. Considering that the number of variables of linear program (3.13) is |E|MU , solvinglinear program (3.13) incurs a time complexity of O(|E|3M3

UL) [94]. As discussed in theproof to Thm. 9, the size of ft which is a solution to our problem is upper bounded by |Eexp|and we have |Eexp| ≤ |E|MU . Those observations, together with the definition of binarysearch, proves that Algorithm 3 has a time complexity of O(|E|3M3

UL logMU).


Proof. Consider one instance in Fig. A.5.

Let us assume d = 7. We have Λ1p = 9, Λ

5/6p = 10, Λ

5/4p = 10 in this instance. Then it is clear

that ΛRp is decreasing with R comparing R1 = 1 with R0 = 5/6, while it is increasing with

R comparing R2 = 5/4 with R1 = 1. Overall, we observe that ΛRp is non-monotonic with R.

114

Besides, we have

Λ0.6R0+0.4R2p = ΛR1

p = 9 < 0.6× 10 + 0.4× 10 = 0.6 · ΛR0p + 0.4 · ΛR2

p ,

implying that ΛRp is non-concave with R. Based on the same instance, we observe that ΛR

a

is non-concave and non-monotonic with R, since ΛR2a = 8.5, ΛR1

a = 7, and ΛR0a = 7.5.

Now let us assume d = 6. We have Λ5/3p = 8, Λ

5/4p = 9, and Λ1

p = 9. Considering R1 = 1,R2 = 5/4, and R3 = 5/3, we have

Λ0.625·R1+0.375·R3p = ΛR2

p = 9 > 0.625× 9 + 0.375× 8 = 0.625 · ΛR1p + 0.375 · ΛR3

p ,

i.e., ΛRp is non-convex with R. Based on the same instance, we observe that ΛR

a is non-convexwith R, since ΛR3

a = 7, ΛR2a = 7.5, and ΛR1

a = 7.


Proof. First, since R is feasible, a feasible periodically repeated solution f = xp(~u) : ∀p ∈P, ∀~u ∈ ~U with T (f) = D/R exists to the problem MPA(Rl, Ru, D) (resp. MAA(Rl, Ru, D)).Based on f we directly construct a feasible solution f = f(R) = xp : ∀p ∈ P to MMD1(R),in order to prove the feasibility of MMD1(R).

Let us set p = p and xp =∑∀~u x

p(~u) · R/D for each p ∈ P . Here we remark that we sendxp data through p without holding them on any node, for each xp of f.

Because f is feasible to MPA(Rl, Ru, D) (resp. MAA(Rl, Ru, D)), the following holds∑∀p

∑∀~u

xp(~u) = D,

implying that∑∀p x

p = R, i.e., the throughput requirement of MMD1(R) is satisfied by f.Next we prove that bandwidth constraints of MMD1(R) is satisfied by f, too.

We note that f must satisfy the bandwidth constraints (3.8), as f is feasible to MPA(Rl, Ru, D)(resp. MAA(Rl, Ru, D))∑

p:e∈p

∑k∈Z,~u:k·T (f)+Bp(~u,e)=i

xp(~u) ≤ be, ∀e ∈ E,∀i = 0, ..., T (f)− 1,

implying the following ∑p:e∈p

∑~u

xp(~u) ≤ T (f) · be, ∀e ∈ E.

115

Because we have defined p = p and xp =∑

~u xp(~u) · R/D for any p ∈ P , we have the

following for any e ∈ E∑p:e∈p

xp =R

D·∑p:e∈p

∑~u

xp(~u) ≤ R

D· T (f) · be = be, where p = p,

i.e., f satisfies the bandwidth constraints of MMD1(R). Therefore, MMD1(R) is feasible,since a feasible solution exists, e.g., f.

Second, let us assume that f(R) is a feasible solution to MMD1(R). Considering that f(R)satisfies bandwidth constraints and sends D amount of data every D/R slots, clearly thatf(R) is feasible to MPA(Rl, Ru, D) (resp. MAA(Rl, Ru, D)) with a throughput of R. Notethat f(R) = xp : ∀p ∈ P is periodically repeated in terms of MPA(Rl, Ru, D) (resp.MAA(Rl, Ru, D)), since it requires that (i) any node except for the sender will not hold anydata at any slot, and (ii) we send xp data to p at each slot, for each path p ∈ P . Accordingto the definition of M(f) [64], in terms of MMD1(R), the following holds

M(f(R)) = maxp∈P :xp>0

∑e∈E:e∈p

de.

As a comparison, in terms of MPA(Rl, Ru, D) (resp. MAA(Rl, Ru, D)), we have

M(f(R)) +D/R− 1 = M(f(R)).

This is because (i) if we consider f(R) as a solution to MMD1(R), it generates R amount ofdata at the sender at each time; if we consider f(R) as a solution to MPA(Rl, Ru, D) (resp.MAA(Rl, Ru, D)), it generates D amount of data generated at the sender every D/R slots.Hence, (ii) if we consider f(R) as a solution to MPA(Rl, Ru, D) (resp. MAA(Rl, Ru, D)), wehave to hold certain data at the sender till the offset D/R − 1, and then send them to thereceiver through paths with delays no greater than M(f(R)).


Proof. Suppose the optimal solution to MMD1(R) is f∗(R).

First, according to Lem. 12, because Rp (resp. Ra) is feasible to MPA(Rl, Ru, D) (resp.MAA(Rl, Ru, D)), we know MMD1(Rp) (resp. MMD1(Ra)) must be feasible, i.e., there is afeasible solution f∗(Rp) (resp. f∗(Ra)). We note that MMD1(Rl) is feasible, since Rl ≤ Rp

(resp. Rl ≤ Ra) implies that we can figure out a solution feasible to MMD1(Rl), after deletingcertain flow rate from f∗(Rp) (resp. f∗(Ra)). Hence we must figure out a feasible solutionfα(Rl), after we use ALG-MMD1(α) to solve MMD1(Rl).

Second, we have that fα(Rl) is a feasible periodically repeated solution to MPA(Rl, Ru, D)(resp. MAA(Rl, Ru, D)) with a throughput of Rl, according to Lem. 12.

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Third, we prove the approximation ratio (α + c) of fα(Rl) with MPA(Rl, Ru, D). Considerthe following

M(fα(Rl))(a)

≤ α · M(f∗(Rl))(b)

≤ α · M(f∗(Rp)), (A.14)

where M(f) is defined in Lem. 12. Inequality (a) comes from the definition of ALG-MMD1(α).Inequality (b) holds since in terms of the min-max-delay flow problem, the minimal maximumdelay is non-decreasing with throughput requirement, considering that given R1 ≥ R2, we canfigure out a solution feasible to MMD1(R2), experiencing a maximum delay lower boundedby f∗(R2), after we delete certain rate from f∗(R1) which is optimal to MMD1(R1).

Let us denote the optimal solution to MPA(Rl, Ru, D) by fp, i.e., fp is feasible with Λp(fp) =

ΛRpp and T (fp) = D/Rp. Note that we can figure out a solution f(Rp) feasible to MMD1(Rp)

from fp follow the proof to Lem. 12. An important observation of f(Rp) is that M(fp) ≥M(f(Rp)), since that during the construction of f(Rp) from fp, (i) f(Rp) does not use anyzero-data path in fp, and (ii) f(Rp) reduces all data-holding delays of fp to zero. We thushave

M(fp) ≥ M(f(Rp))(a)

≥ M(f∗(Rp))(b)

≥ M(fα(Rl))

α

(c)=M(fα(Rl))−D/Rl + 1

α, (A.15)

where inequality (a) holds due to the optimality of f∗(Rp) as to MMD1(Rp), the inequality(b) comes from the inequality (A.14), and the equality (c) holds due to our Lem. 12.

Now according to Lem. 4 and the inequality (A.15), we have

Λp(fα(Rl))

Λp(fp)=M(fα(Rl)) +D/Rl − 1

M(fp) +D/Rp − 1≤ α · M(fp) + 2D/Rl − 2

M(fp) +D/Rp − 1

=α · M(fp)

M(fp) +D/Rp − 1+

2(D/Rl − 1)

M(fp) +D/Rp − 1(a)

≤ α +2(D/Rl − 1)

D/Rp

≤ α + 2 · Ru

Rl

,

where the inequality (a) is true because thatM(fp) ≥ 1 and D/Rp ≥ 1 since de ∈ Z+ for eache ∈ E. Overall, the approximation ratio (α+ c) of fα(Rl) holds in terms of MPA(Rl, Ru, D).

Finally, we can prove the approximation ratio (α+c) of fα(Rl) with MAA(Rl, Ru, D) similarly.


Proof. First, after relaxing maximum delay constraints of MUDT, we consider the problemas follows

max −M(f1)

117

s.t. |f1| ≥ R1,

M(f1) ≤ +∞,f = f1 ∈ X .

The problem above has been proven to be NP-complete [64].

Second, after relaxing throughput requirements of MUDT, we consider the problem below

max |f1|s.t. |f1| ≥ 0,

M(f1) ≤ D1,

f = f1 ∈ X .

The above problem can be proven to be NP-complete, similarly as discussed in the Appendixof [64].

Third, consider the problem below that is a special case of MUDT. It can be proven that it isNP-complete even to generate a solution feasible to it with all constraints satisfied, similarlyas discussed in the Appendix of [64],

max U t1(|f1|)s.t. |f1| ≥ R1,

M(f1) ≤ D1,

f = f1 ∈ X ,

where U t1(|f1|) = 1 which is a constant.


Proof. First, problem (4.2) and (4.3) both can be solved in polynomial time because of thecondition 1, as (i) they are convex programs with a polynomial size, and (ii) it takes apolynomial time to solve such programs up to an arbitrarily small additive error [28,68]. Weobtain K edge-defined single-unicast flows after solving the average-delay-aware problem.Then we can figure out K path-defined single-unicast flows f = fi, i = 1, 2, ..., K in a timeof O(|V |2|E|K), using the classic flow decomposition technique [24]. Note that for each fi,the number of paths given by flow decomposition is upper bounded by |E|, implying thatobtaining fi by deleting rate from fi terminates in at most |E| iterations. Overall, we observethat the time complexity of Algorithm 4 is polynomial.

Second, we prove f must exist.

(i) If the optimization objective is (4.1a), f ∗ satisfies all constraints of problem (4.1), sincef ∗ is the optimal solution to the feasible problem (4.1), implying that f ∗ also satisfies the

118

constraints (4.2b) and (4.2d) of problem (4.2). Now for any i = 1, 2, ..., K, the followingholds since T (g) ≤M(g) · |g| holds for any single-unicast flow g,

T (f ∗i ) ≤ M(f ∗i ) · |f ∗i |(a)

≤ Di · |f ∗i |,

where the inequality (a) comes from the satisfied constraints (4.1d). f ∗ is thus feasible tothe problem (4.2), implying that problem (4.2) is feasible and hence Algorithm 4 gives asolution f .

(ii) If the optimization objective is (4.1b), f ∗ meets all constraints of problem (4.1), as f ∗

is an optimal solution to problem (4.1). Considering that |f ∗i | ≥ Ri,∀i = 1, 2, ..., K, we canconstruct another flow f as follows from f ∗: we obtain fi from f ∗i by deleting rate of f ∗itill |f ∗i | = Ri, for each i = 1, 2, ..., K. It is clear that such a solution f must exist, withthroughput requirements (4.3b) satisfied. The satisfied constraint (4.1e) of f ∗ implies thatf meets constraint (4.3d). Since the maximum delay will not increase during the process ofdeleting rate, the following holds

M(fi) ≤ M(f ∗i ), ∀i = 1, 2, ..., K, (A.17)

implying for any i = 1, 2, ..., K we have

T (fi) ≤ M(fi) · |fi| = M(fi) ·Ri ≤ M(f ∗i ) ·Ri ≤ Di ·Ri,

i.e., f satisfies the constraints (4.3c). f is thus feasible to the problem (4.3), implying thatAlgorithm 4 must give a solution f .

Third, we prove that f meets constraints (4.5). Let us denote the solution given in line 5 byf . It is clear that we have ∣∣∣fi∣∣∣ ≥ Ri, ∀i = 1, 2, ..., K, (A.18a)

A(fi

)≤ Di, ∀i = 1, 2, ..., K, (A.18b)

f = f1, f2, ..., fK ∈ X . (A.18c)

Since for each i = 1, 2, ..., K, fi deletes a rate of ε · |fi| from fi, we know that f meetsconstraints (4.5a) and (4.5c).

According to Lem. 13, we have

ε ·∣∣∣fi∣∣∣ · M (

fi)≤ T

(fi

)− T

(fi)≤ T

(fi

), ∀i = 1, 2, ..., K,

implying that M(fi) ≤ A(fi)/ε,∀i = 1, 2, ..., K. Based on the satisfied constraints (A.18b),it holds that

M(fi) ≤ A(fi)/ε ≤ Di/ε, ∀i = 1, 2, ..., K.

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Finally, we prove the approximation ratio of f . The following holds if the optimizationobjective isf (4.1a)

K∑i=1

U ti(∣∣fi∣∣) =

K∑i=1

U ti(

(1− ε) ·∣∣∣fi∣∣∣) (a)

≥ (1− ε) ·K∑i=1

U ti(∣∣∣fi∣∣∣) (b)

≥ (1− ε) ·K∑i=1

U ti (|f ∗i |)

where the inequality in (b) holds because f is optimal to the problem (4.2), while f ∗ isfeasible it (see the second part of this proof). For each i = 1, 2, ..., K, the inequality in (a)holds due to the following

U ti(

(1− ε) ·∣∣∣fi∣∣∣) = U ti

(ε · 0 + (1− ε) ·

∣∣∣fi∣∣∣) (c)

≥ ε·U ti (0)+(1−ε)·U ti(∣∣∣fi∣∣∣) (d)

≥ (1−ε)·U ti(∣∣∣fi∣∣∣) ,

where the inequality (c) comes from the concavity of U ti (·), and the inequality (d) holds dueto its non-negativity.

If the optimization objective is (4.1b), let us denote f as the solution feasible to the prob-lem (4.3) and is constructed from f ∗. It holds that

K∑i=1

Udi(M(fi)

)≤

K∑i=1

Udi(A(fi

)/ε) (a)

≤ 1

ε·K∑i=1

Udi(A(fi

)) (b)

≤ 1

ε·K∑i=1

Udi (A (fi))

≤ 1

ε·K∑i=1

Udi (M (fi))(c)

≤ 1

ε·K∑i=1

Udi (M(f ∗i )),

where the inequality (a) holds due to the satisfied condition 2, the inequality (b) is true sincef is optimal to problem (4.3) but f is only feasible it, and the inequality (c) comes from theinequality (A.17).


Proof. First, as proved in Thm. 18, the time complexity of Algorithm 5 is polynomial, andf must exist.

Second, constraints (4.8b) and (4.8c) holds straightforwardly. We define (|fi| − |fi|)/|fi| asεi. Thus εmin ≤ εi ≤ εmax for any i = 1, 2, ..., K, implying∣∣fi∣∣ = (1− εi) ·

∣∣∣fi∣∣∣ ≥ (1− εmax) ·∣∣∣fi∣∣∣ , ∀i = 1, 2, ..., K,

i.e., the constraints (4.8b) are met.

Third, we can prove the approximation ratio (4.9) by following the same proof as to Thm. 18.

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As for the approximation ratio (4.10), we assume that after we delete εmin|fi| rate from theslowest flow-carrying paths of fi for each i = 1, 2, ..., K, we can obtain a solution fi. Thenit holds that

M(fi) ≤ M(fi), ∀i = 1, 2, ..., K,

because for each i = 1, 2, ..., K, the amount of deleted rate to obtain fi is lower bounded bythat to obtain fi. We thus have

K∑i=1

Udi(M(fi))≤

K∑i=1

Udi(M(fi

)) (a)

≤ 1

εmin

·K∑i=1

Udi (M (f ∗i )) ,

where the inequality (a) holds due to Thm. 18, as f is the returned solution when usingAlgorithm 4 to solve problem (4.1) with ε = εmin.


Proof. Same to Thm. 18, we can prove that the time complexity of PASS-T is polynomial,and PASS-T must give a solution f , satisfying constraints (4.11a) and (4.11c), achieving theapproximation ratio (4.12).

Considering that g is the solution of PASS, it holds that

M (gi) ≤ Di/ε, ∀i = 1, 2, ..., K, (A.19)K∑i=1

Udi (M (gi)) ≤1

ε·K∑i=1

Udi (M (f ∗i )) . (A.20)

By the definition of λ, we have

M(fi) ≤ λ · M(gi), ∀i = 1, 2, ..., K,

implying the followingM(fi) ≤ λ ·Di/ε, ∀i = 1, 2, ..., K,

i.e., the constraints (4.11b) are satisfied. Moreover, the following holds

K∑i=1

Udi(M(fi)

)≤

K∑i=1

Udi (λ · M(gi))(a)

≤ λ ·K∑i=1

Udi (M(gi))(b)

≤ λ

ε·K∑i=1

Udi (M(f ∗i )) ,

where the inequality (a) comes from condition 2, and the inequality (b) comes from theinequality (A.20). Thus the approximation ratio (4.13) holds.

121


Proof. We observe that both formulations share the same objective function. Hence we onlyneed to prove that a solution ~xi,~ti, i = 1, 2, ..., K is feasible with the formulation in (5.2)if and only if it is feasible with the formulation in (5.4),

If part. Suppose ~xi,~ti, i = 1, 2, ..., K meets the constraint (5.2b), we prove it must meetthe constraint (5.4b).

For such a solution ~xi,~ti, i = 1, 2, ..., K, given a specific i ∈ 1, 2, ....K, if ai−1 ≤ T outi , then

the following holds

max ai−1, Touti +

∑e∈E

xei tei = T out

i +∑e∈E

xei tei .

Based on the constraint (5.2b) when r = k = i, clearly it holds

max ai−1, Touti +

∑e∈E

xei tei ≤ T out

i + T ini+1 − T out

i = T ini+1,

which is the constraint (5.4b).

If ai−1 > T outi , it is fair to assume that aj > T out

j+1 for j = l, l + 1, ..., i − 1 where l ≥ 1 andal−1 ≤ T out

l . Then it holds that

maxai−1, Touti +

∑e∈E

xei tei = ai−1 +

∑e∈E

xei tei = ai−2 +

∑e∈E

xei−1tei−1 +

∑e∈E

xei tei

= ... = al +i∑

j=l+1

∑e∈E

xejtej = T out

l +i∑j=l

∑e∈E

xejtej .

Based on the constraint (5.2b) when r = i, k = l, we have

max ai−1, Touti +

∑e∈E

xei tei ≤ T out

l + T ini+1 − T out

l = T ini+1,


Only if part. Suppose ~xi,~ti, i = 1, 2, ..., K meets the constraint (5.4b). Now we prove itmust meet the constraint (5.2b).

Due to the assumption that ai = maxai−1, Touti +

∑e∈E x

ei tei and ai ≤ T in

i+1, considering thetwo inequalities maxai−1, T

outi ≥ ai−1 and maxai−1, T

outi ≥ T out

i , for any r = 1, 2, ..., Kand any k = 1, 2, ..., r, the following holds∑e∈E

xerter ≤ T in

r+1 −max ar−1, Toutr ≤ T in

r+1 − ar−1 = T inr+1 −max

ar−2, T

outr−1

−∑e∈E

xer−1ter−1

≤ T inr+1 −

∑e∈E

xer−1ter−1 − ar−2 = T in

r+1 −r−1∑j=r−2

∑e∈E

xejtej −max

ar−3, T

outr−2

122

≤ ... = T inr+1 −

r−1∑j=k

∑e∈E

xejtej −max ak−1, T

outk ≤ T in

r+1 −r−1∑j=k

∑e∈E

xejtej − T out

k ,

namely it holds thatr∑j=k

xejtej ≤ T in

r+1 − T outk ,



Proof. The proof follows the following equalities

D(~λ) = min~x,~t

L(~x,~t, ~λ) = −K∑r=1

r∑k=1

λrk(Tinr+1 − T out

k ) + min~x,~t

K∑i=1

∑e∈E

xei · [ce(tei , ρi) + µitei ]

(a)= −

K∑r=1

r∑k=1


k ) +K∑i=1

min~xi

min~ti

∑e∈E

xei · [ce(tei , ρi) + µitei ]

(b)= −

K∑r=1

r∑k=1


k ) +K∑i=1

min~xi

∑e∈E

xei · mintle≤tei≤te(ρi)

[ce(tei , ρi) + µit

ei ]

(c)= −

K∑r=1

r∑k=1


k ) +K∑i=1

min~xi

∑e∈E

xei · [ce(tei (µi), ρi) + µitei (µi)]

(d)= −

K∑r=1

r∑k=1


k ) +K∑i=1

min~xi

∑e∈E

xei · wei (µi)

(e)= −

K∑r=1

r∑k=1


k ) +K∑i=1

∑e∈p(µi)

wei (µi),

where (a) is true because no coupled constraints exist for different tasks and no coupledconstraints exist between ~xi and ~ti given any i = 1, 2, ..., K. (b) holds since for the traveltime on different edges given any i = 1, 2, ..., K, no coupled constraints exist. (c) comes fromour definition of tei (µi). Similarly, (d) comes from the definition of wei (µi), and (e) comesfrom the definition of p(µi).


Proof. First we prove the solution p(µi), i = 1, 2, ..., K that satisfies the condition (5.10)must be feasible to MEET.

123

It is clear that for each task τi, i = 1, 2, ..., K, such a solution defines a path from σi to σi+1

to fulfill τi. We need to prove that it meets all the time window constraint (5.4b), in orderto prove its feasibility. For any r = 1, 2, ..., K and k = 1, 2, ..., r, if the corresponding λrk isstrictly positive, clearly based on the condition (5.10) we have

T outk − T in

r+1 +r∑j=k

δ(µj) = 0,

implying that the solution p(µi), i = 1, 2, ..., K satisfies the constraint (5.4b). For anyr = 1, 2, ..., K and k = 1, 2, ..., r, if the corresponding λrk = 0, based on the condition (5.10)the following holds

max

T outk − T in

r+1 +r∑j=k

δ(µj), 0

= 0,

namely we have

T outk − T in

r+1 +r∑j=k

δ(µj) ≤ 0,

also implying that the solution p(µi), i = 1, 2, ..., K satisfies the constraint (5.4b). Overall,the solution p(µi), i = 1, 2, ..., K is feasible to MEET. Therefore, its cost upper bounds theoptimal cost of MEET, i.e.

P(~λ) =K∑i=1

∑e∈p(µi)

ce(tei (µi), ρi) ≥ OPT.

Next we look at the dual function of p(µi), i = 1, 2, ..., K.

D(~λ) = −K∑r=1

r∑k=1


k ) +K∑i=1

∑e∈p(µi)

[ce(tei (µi), ρi) + µit

ei (µi)]

= −K∑r=1

r∑k=1


k ) +K∑i=1

µiδ(µi) +K∑i=1

∑e∈p(µi)

ce(tei (µi), ρi)

=K∑r=1

r∑k=1

λrk ·

(T outk − T in

r+1 +r∑j=k

δ(µj)

)+

K∑i=1

∑e∈p(µi)

ce (tei (µi), ρi) .

Since the condition (5.10) implies that

λrk ·

(T outk − T in

r+1 +r∑j=k

δ(µj)

)= 0, ∀r = 1, 2, ..., K,∀k = 1, 2, ..., r,

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the dual function is

D(~λ) =K∑i=1

∑e∈p(µi)

ce(tei (µi), ρi).

According to the weak duality, we have the following

D(~λ) =K∑i=1

∑e∈p(µi)

ce(tei (µi), ρi) ≤ OPT.

Since we have D(~λ) ≤ OPT ≤ P(~λ) and D(~λ) = P(~λ), it is clear that the solution p(µi), i =1, 2, ..., K must be optimal to MEET.


Proof. If MEET is feasible, the solution where each task is fulfilled by the fastest path mustbe feasible, which must be examined by SPEED due to that initially we set all dual variablesto be a large enough λmax. Therefore, SPEED must return a feasible solution.

Result for the first case holds directly due to Thm. 25.

For p returned in the second case, let us denote the dual variables defining p as λrk, ∀r =

1, 2, ..., K,∀k = 1, 2, ..., r, and denote max∀r,k | ˙λrk| as ˙λ, namely | ˙λrk| ≤ ˙λ,∀r = 1, 2, ..., K,∀k =1, 2, ..., r.

Because p is feasible, we have

P(~λ) =K∑i=1

∑e∈p(µi)

ce (tei (µi), ρi) ≥ OPT.

Next we look at the dual function of p.

D(~λ) = −K∑r=1

r∑k=1


k ) +K∑i=1

µiδ(µi) +K∑i=1

∑e∈p(µi)

ce (tei (µi), ρi)

=K∑r=1

r∑k=1

λrk ·

(T outk − T in

r+1 +r∑j=k

δ(µj)

)+

K∑i=1

∑e∈p(µi)

ce (tei (µi), ρi) .

Since p is feasible, we have

r∑j=k

δ(µj) ≤ T inr+1 − T out

k ,∀r = 1, 2, ..., K,∀k = 1, 2, ..., r,

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implying the following holds[T outk − T in

r+1 +r∑j=k

δ(µj)

]+

λrk

≤ 0.

Further due to φ(λrk) > 0, it holds that ˙λrk ≤ 0, implying that − ˙λ ≤ ˙λrk ≤ 0, and we have[T outk − T in

r+1 +r∑j=k

δ(µj)

]+

λrk

≥ − ˙λ/φ(λrk).

Considering the definition of [T outk − T in

r+1 +∑r

j=k δ(µj)]+λrk

, we have

λrk

(T outk − T in

r+1 +r∑j=k

δ(µj)

)= λrk

[T outk − T in

r+1 +r∑j=k

δ(µj)

]+

λrk

≥ − ˙λ · λrk/φ(λrk).

According to the weak duality, we have

D(~λ) ≤ OPT.

Thus the following holds

P(~λ)− OPT ≤ P(~λ)−D(~λ) = −K∑r=1

r∑k=1

λrk ·

(T outk − T in

r+1 +r∑j=k

δ(µj)

)≤ K2 · ˙λ ·max

∀r,k

λrk/φ(λrk)

= K2 ·max

∀r,k

∣∣∣ ˙λrk∣∣∣ ·max∀r,k

λrk/φ(λrk)

.

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