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Facility Location with Client Latencies: Linear-Programming basedTechniques for Minimum-Latency Problems

Deeparnab Chakrabarty∗ Chaitanya Swamy†

Abstract

We introduce a problem that is a common generalization of theuncapacitated facility location(UFL)andminimum latency(ML) problems, where facilities not only need to be opened to serve clients, but alsoneed to be sequentially activated before they can provide service. This abstracts a setting where inventorydemanded by customers needs to be stocked or replenished at facilities from a depot or warehouse.Formally, we are given a setF of n facilities with facility-opening costsfi, a setD of m clients, andconnection costscij specifying the cost of assigning a clientj to a facility i, a root noder denoting thedepot, and atime metricd onF ∪ r. Our goal is to open a subsetF of facilities, find a pathP startingat r and spanningF to activate the open facilities, and connecting each clientj to a facilityφ(j) ∈ F ,so as to minimize

∑

i∈F fi +∑

j∈D(cφ(j),j + tj), wheretj is the time taken to reachφ(j) along pathP . We call this theminimum latency uncapacitated facility location(MLUFL) problem.

Our main result is anO(

lognmaxlogn, logm)

-approximation forMLUFL. Via a reduction to thegroup Steiner tree (GST) problem, we show this result is tight in the sense that any improvement in theapproximation guarantee forMLUFL, implies an improvement in the (currently known) approximationfactor for GST. We obtain significantly improved constant approximation guarantees for two naturalspecial cases of the problem: (a)relatedMLUFL, where the connection costs form a metric that is ascalar multiple of the time metric; (b)metric uniformMLUFL, where we have metric connection costsand the time-metric is uniform. Our LP-based methods are fairly versatile and are easily adapted withminor changes to yield approximation guarantees forMLUFL (andML) in various more general settings,such as (i) the setting where the latency-cost of a client is afunction (of bounded growth) of the delayfaced by the facility to which it is connected; and (ii) thek-route version, where we can dispatchkvehicles in parallel to activate the open facilities.

Our LP-based understanding ofMLUFL also offers some LP-based insights intoML. We obtain twonatural LP-relaxations forML with constant integrality gap, which we believe shed new light upon theproblem and offer a promising direction for obtaining improvements forML.

1 Introduction

Facility location and vehicle routing problems are two broad classes of combinatorial optimization problemsthat have been widely studied in the Operations Research community (see, e.g., [25, 32]), and have a widerange of applications. Both problems can be described in terms of an underlying set of clients that need to beserviced. In facility location problems, there is a candidate set of facilities that provide service, and the goalis to open some facilities and connect each client to an open facility so as to minimize some combinationof the facility-opening and client-connection costs. Vehicle routing problems consider the setting where avehicle (delivery-man or repairman) provides service, andthe goal is to plan a route that visits (and henceservices) the clients as quickly as possible. Two common objectives considered are: (i) minimize the total

∗deepc@seas.upenn.edu. Dept. of Computer and Information Science, Univ. of Pennsylvania, Philadelphia, PA 19104.

Most of the work was done as a postdoctoral fellow at the Dept.of Combinatorics and Optimization, Univ. Waterloo.†cswamy@math.uwaterloo.ca. Dept. of Combinatorics and Optimization, Univ. Waterloo,Waterloo, ON N2L 3G1.

Supported in part by NSERC grant 327620-09 and an Ontario Early Researcher Award.

1

length of the vehicle’s route, giving rise to the traveling salesman problem (TSP), and (ii) (adopting a client-oriented approach) minimize the sum of the client delays, giving rise to minimum latency (ML) problems.

These two classes of problems have mostly been considered separately. However, various logisticsproblems involve both facility-location and vehicle-routing components. For example, consider the follow-ing oft-cited prototypical example of a facility location problem: a company wants to determine where toopen its retail outlets so as to serve its customers effectively. Now, inventory at the outlets needs to bereplenished or ordered (e.g., from a depot); naturally, a customer cannot be served by an outlet unless theoutlet has the inventory demanded by it, and delays incurredin procuring inventory might adversely impactcustomers. Hence, it makes sense for the company toalsokeep in mind the latencies faced by the customerswhile making its decisions about where to open outlets, how to connect customers to open outlets, and inwhat order to replenish the open outlets, thereby adding a vehicle-routing component to the problem.

We propose a mathematical model that is a common generalization of theuncapacitated facility location(UFL) andminimum latency(ML) problems, and abstracts a setting (such as above) where facilities need tobe “activated” before they can provide service. Formally, as inUFL, we have a setF of n facilities, and asetD of m clients. Opening facilityi incurs afacility-opening costfi, and assigning a clientj to a facilityi incursconnection costcij . Taking a lead from minimum latency problems, we model activation delays asfollows. We have a root (depot) noder, and atime metricd onF ∪ r. A feasible solution specifies asubsetF ⊆ F of facilities to open, a pathP starting atr and spanningF along which the open facilities areactivated, and assigns each clientj to an open facilityφ(j) ∈ F . The cost of such a solution is

∑

i∈F

fi +∑

j∈D

(

cφ(j)j + tj)

(1)

wheretj = dP (r, φ(j)) is the time taken to reach facilityφ(j) along pathP . We refer totj as clientj’slatency cost. The goal is to find a solution with minimum totalcost. We call this theminimum-latencyuncapacitated facility location(MLUFL) problem.

Apart from being a natural problem of interest, we findMLUFL appealing since it generalizes, or isclosely-related to, various diverse problems of interest (in addition toUFL andML); our work yields newinsights on some of these problems, most notablyML (see “Our results”). One such problem, which capturesmuch of the combinatorial core ofMLUFL is what we call theminimum group latency(MGL) problem. Here,we are given an undirected graph with metric edge weightsde, subsetsGj of vertices called groups,and a rootr; the goal is to find a path starting atr that minimizes the sum of the cover times of the groups,where the cover time ofGj is the first time at which somei ∈ Gj is visited on the path. Observe thatMGLcan be cast asMLUFL with zero facility costs (whereF = node-set\ r), where for each groupGj , wecreate a clientj with cij = 0 if i ∈ Gj and∞ otherwise. Note that we may assume that the groups aredisjoint (by creating multiple co-located copies of a node), in which case thesecijs form a metric.MGLitself captures various other problems. Clearly, when eachGj is a singleton, we obtain the minimum latencyproblem. Also, given a set-cover instance, if we consider a graph whose nodes are (r and) the sets, create agroupGj for each elementj consisting of the sets containing it, and consider the uniform metric, then thisMGL problem is simply themin-sum set cover(MSSC) problem [16].

Our results and techniques. Our main result is anO(

log nmaxlogm, log n)

-approximation algorithmfor MLUFL (Section 2.1), which for the special case ofMGL, implies anO(log2 n) approximation. Com-plementing this result, we prove (Theorem 2.9) that aρ-approximation algorithm (even) forMGL yieldsan O(ρ logm)-approximation algorithm for thegroup Steiner tree(GST) problem [17] onn nodes andm groups. Thus, any improvement in our approximation ratio for MLUFL would yield a correspondingimprovement ofGST, whose approximation ratio has remained atO(log2 n logm) for a decade [17]. More-over, combined with the result of [22] on the inapproximability of GST, this shows thatMGL, and henceMLUFL with metric connection costs cannot be approximated to better than aΩ(logm)-factor unlessNP⊆ZTIME (npolylog(n)).

2

Given the above hardness result, we investigate certain well-motivated special cases ofMLUFL andobtain significantly improved performance guarantees. In Section 2.2, we consider the case where theconnection costs form a metric, which is a scalar multiple ofthe d-metric (i.e.,duv = cuv/M , whereM ≥ 1; the problem is trivial ifM < 1). For example, in a supply-chain logistics problem, this models anatural setting where the connection of clients to facilities, and the activation of facilities both proceed alongthe same transportation network. We obtain aconstant-factorapproximation algorithm for this problem.

In Section 2.3, we consider theuniform MLUFL problem, which is the special case where the time-metric is uniform. UniformMLUFL already generalizesMSSC (and alsoUFL). For uniformMLUFL withmetric connection costs (i.e., metric uniformMLUFL), we devise a10.78-approximation algorithm. (Withoutmetricity, the problem becomes set-cover hard, and we obtain a simple matchingO(logm)-approximation.)The chief novelty here lies in the technique used to obtain this result. We give a simple generic reduction(Theorem 2.12) that shows how to reduce the metric uniformMLUFL problem with facility costs to onewithout facility costs, in conjunction with an algorithm forUFL. This reduction is surprisingly robust andversatile and has other applications. For example, the samereduction yields anO(1)-approximation formetric uniformk-median(i.e., metric uniformMLUFL where at mostk facilities may be opened), and thesame ideas lead to improved guarantees for thek-median versions of connected facility location [31], andfacility location with service installation costs [28].

We obtain our approximation bounds by rounding the optimal solution to a suitable linear-programming(LP) relaxation of the problem. This is interesting since weare not aware of any previous LP-based methodsto attackML (as a whole). In Section 3, we leverage this to obtain some interesting insights aboutML, whichwe believe cast new light on the problem. In particular, we present two LP-relaxations forML, and prove thatthese have (small) constant integrality gap. Our first LP is aspecialization of our LP-relaxation forMLUFL.Interestingly, the integrality-gap bound for this LP relies only on the fact that the natural LP relaxation forTSP has constant integrality gap (i.e., aρ-integrality gap for the natural TSP LP relaxation translates to anO(ρ)-integrality gap). In contrast, the various known algorithms forML [7, 10, 1] all utilize algorithms forthe arguably harderk-MST problem or its variants. Our second LP has exponentially-many variables, onefor every path (or tree) of a given length bound, and the separation oracle for the dual problem is a rooted path(or tree) orienteering problem: given rewards on the nodes and metric edge costs, find a (simple) path rootedat r of length at mostB that gathers maximum reward. We prove that even a bicriteriaapproximation forthe orienteering problem yields an approximation forML while losing a constant factor. This connectionbetween orienteering andML is known [14]. But we feel that our alternate proof, where theorienteeringproblem appears as the separation oracle required to solve the dual LP, offers a more illuminating explanationof the relation between the approximability of the two problems. (In fact, the same relationship also holdsbetweenMGL and “group orienteering.”)

We believe that the use of LPs opens upML to new venues of attack. A good LP-relaxation is beneficialbecause it yields a concrete, tractable lower bound and handle on the integer optimum, which one canexploit to design algorithms (a point repeatedly driven home in the field of approximation algorithms). Also,LP-based techniques tend to be fairly versatile and can be adapted to handle more general variants of theproblem (more on this below). Our LP-rounding algorithms exploit various ideas developed for schedulingand facility-location problems (e.g.,α-points) and polyhedral insights forTSP, which suggests that thewealth of LP-based machinery developed for these problems can be leveraged to obtain improvements forML. We suspect that our LP-relaxations are in fact better than what we have accounted for, and considerthem to be a promising direction for making progress onML.

Section 4 showcases the flexibility afforded by our LP-basedtechniques, by showing that our algorithmsand analyses extend with little effort to handle various generalizations ofMLUFL (and hence,ML). For exam-ple, consider the setting where the latency-cost of a clientj is λ(time taken to reach the facility servingj),for some non-decreasing functionλ(.). Whenλ is convex and has “growth” at mostp (i.e.,λ(cx) ≤ cpλ(x)),we derive anO

(

max(p log2 n)p, p log n logm)

-approximation forMLUFL, anO(

2O(p))

-approximation

3

for relatedMLUFL andML, and anO(1)-approximation for metric uniformMLUFL. (Concaveλs are eveneasier to handle.) This in turn leads to approximation guarantees for theLp-norm generalization ofMLUFL,where instead of the sum (i.e.,L1-norm) of client latencies, theLp-norm of the client latencies appears inthe objective function. The spectrum ofLp norms tradeoff efficiency with fairness, making theLp-normproblem an appealing problem to consider. We obtain anO

(

p log nmaxlog n, logm)

-approximation forMLUFL, and anO(1)-approximation for the other special cases. Another notable extension is thek-routeversion of the problem, where we may usek paths starting atr to traverse the open facilities. With one sim-ple modification to our LPs (and algorithms),all our approximation guarantees translate to this setting. Asacorollary, we obtain a constant-factor approximation for theLp-norm version of thek-traveling repairmenproblem[14] (thek-route version ofML).

Related work. To the best of our knowledge,MLUFL and MGL are new problems that have not beenstudied previously. There is a great deal of literature on facility location and vehicle routing problems (see,e.g., [25, 32]) in general, andUFL andML, in particular, which are special cases of our problem, and welimit ourselves to a sampling of some of the relevant results. The first constant approximation guaranteefor UFL was obtained by Shmoys, Tardos, and Aardal [29] via an LP-rounding algorithm, and the currentstate-of-the-art is 1.5-approximation algorithm due to Byrka [8]. The minimum latency (ML) problem seemsto have been first introduced to the computer science community by Blum et al. [7], who gave a constant-factor approximation algorithm for it. Goemans and Kleinberg [19]improved the approximation factor to10.78, using a “tour-concatenation” lemma, which has formed a component of all subsequent algorithmsand improvements forML. The current-best approximation factor forML is 3.59 due to Chaudhuri, Godfrey,Rao and Talwar [10]. As mentioned earlier,MGL with a uniform time metric captures the min-sum setcover (MSSC) problem. This problem was introduced by Feige, Lovasz and Tetali [16], who gave a4-approximation algorithm for the problem and a matching inapproximability result. Recently, Azar et al. [2]introduced a generalization ofMSSC, for which Bansal et al. [5] obtained a constant-factor approximation.

If instead of adding up the latency cost of clients, we include themaximumlatency cost of a client inthe objective function ofMLUFL, then we obtain themin-max versions ofMGL andMLUFL, which havebeen studied previously. Themin-max version ofMGL is equivalent to a “path-variant” ofGST: we seek apath starting atr of minimum total length that covers every group. Garg, Konjevod, and Ravi [17] devisedan LP-rounding basedO(log2 n logm)-approximation forGST, wheren is the number of nodes andmis the number of groups. Charikar et al. [9] gave a deterministic algorithm with the same guarantee, andHalperin and Krauthgamer [22] proved anΩ(log2 m)-inapproximability result. The rounding techniqueof [17] and the deterministic tree-embedding constructionof [9] (rather its improvement by [15]) are keyingredients of our algorithm for (general)MLUFL. Themin-max version ofMLUFL can be viewed a path-variant of connected facility location [27, 20]. The connected facility location problem, in its full generality,is essentially equivalent toGST [27]; however, if the connection costs form a metric, and thetime- and theconnection-cost metrics are scalar multiples of each other, then various constant-factor approximations areknown [20, 31, 13].

Very recently, we have learnt that, independent of, and concurrent with, our work, Gupta et al. [21] alsopropose the minimum group latency (MGL) problem (which they arrive at in the course of solving a differentproblem), and obtain results similar to ours forMGL. They also obtain anO(log2 n)-approximation forMGL,and the reduction fromGST to MGL with a logm-factor loss (see also [26]), and relate the approximabilityof theMGL and “group orienteering” problems. Their techniques are combinatorial and not LP-based, andit is not clear how these can be extended to handle facility-opening costs.

4

2 LP-rounding approximation algorithms for MLUFL

We can expressMLUFL as an integer program and relax the integrality constraintsto obtain a linear programas follows. We may assume thatdii′ is integral for alli, i′ ∈ F ∪r. LetE denote the edge-set of the com-plete graph onF ∪ r and letdmax := maxe∈E de. LetT ≤ minn,mdmax be a known upper bound onthe maximum activation time of an open facility in an optimalsolution. For every facilityi, clientj, and timet ≤ T, we have a variableyi,t indicating if facility i is opened at timet or not, and a variablexij,t indicatingwhether clientj connects to facilityi at timet. Also, for every edgee ∈ E and timet, we introduce a vari-ableze,t which denotes if edgee has been traversedby time t. Throughout, we usei to index the facilities inF , j to index the clients inD, t to index the time units in[T] := 1, . . . ,T, ande to index the edges inE.

min∑

i,t

fiyi,t +∑

j,i,t

(

cij + t)

xij,t (P)

s.t.∑

i,t

xij,t ≥ 1 for all j; xij,t ≤ yi,t for all i, j, t

∑

e

deze,t ≤ t for all t (2)

∑

e∈δ(S)

ze,t ≥∑

i∈S,t′≤t

xij,t′ for all t, S ⊆ F , j (3)

xij,t, yi,t, ze,t ≥ 0 for all i, j, t, e; yi,t = 0 for all i, t with dir > t.

The first two constraints encode that each client is connected to some facility at some time, and that if aclient is connected to a facilityi at timet, theni must be open at timet. Constraint (2) ensures that by timet no more thant “distance” is covered by the tour on facilities, and (3) ensures that if a client is connectedto i by timet, then the tour must have visitedi by timet. We assume for now thatT = poly(m), and showlater how to remove this assumption (Lemma 2.7, Theorem 2.8). Thus, (P) can be solved efficiently sinceone can efficiently separate over the constraints (3). Let(x, y, z) be an optimal solution to (P), andOPT

denote its objective value. For a clientj, defineC∗j =

∑

i,t cijxij,t, andL∗j =

∑

i,t txij,t. We devise variousapproximation algorithms forMLUFL by rounding(x, y, z) to an integer solution.

In Section 2.1, we give a polylogarithmic approximation algorithm for (general)MLUFL (where thecijsneed not even form a metric). Complementing this result, we prove (Theorem 2.9) that aρ-approximationalgorithm (not necessarily LP-based) forMLUFL yields anO(ρ logm)-approximation algorithm for theGSTproblem onn nodes andm groups. In Sections 2.2 and 2.3, we obtain significantly-improved approximationguarantees for various well-motivated special cases ofMLUFL. Section 2.2 obtains aconstant-factorapprox-imation algorithm in the natural setting where the connection costs form a metric that is a scalar multipleof the time-metric. Section 2.3 considers the setting wherethe time-metric is the uniform metric. Our mainresult here is a constant-factor approximation for metric connection costs, which is obtained via a ratherversatile reduction of this uniformMLUFL problem toUFL and uniformMLUFL with zero-facility costs.

2.1 An O(

logn · maxlogn, logm)

-approximation algorithm

We first give an overview. LetNj = i ∈ F : cij ≤ 4C∗j be the set of facilities “close” toj, and defineτj

as the earliest timet such that∑

i∈Nj ,t′≤t xij,t′ ≥23 . By Markov’s inequality, we have

∑

i∈Nj

∑

t xij,t ≥34

andτj ≤ 12L∗j . It is easiest to describe the algorithm assuming first that the time-metricd is a tree metric.

Our algorithm runs in phases, with phaseℓ corresponding to timetℓ = 2ℓ. In each phase, we compute arandom subtree rooted atr of “low” cost such that for every clientj with τj ≤ tℓ, with constant probability,this tree contains a facility inNj . To compute this tree, we utilize the rounding procedure of Garg-Konjevod-Ravi (GKR) for thegroup Steiner tree(GST) problem [17] (see Lemma 2.4), by creating a group for each

5

client j with τj ≤ tℓ comprising of, roughly speaking, the facilities inNj. We open all the facilities includedin the subtree, and obtain a tour via the standard trick of doubling all edges and performing an Eulerian tourwith possible shortcutting. The overall tour is a concatenation of all the tours obtained in the various phases.For each clientj, we consider the first tree that contains a facility fromNj (which must therefore be open),and connectj to such a facility.

Given the result for tree metrics, an oft-used idea to handlethe case whend is not a tree metric is toapproximate it by a distribution of tree metrics withO(log n) distortion [15]. Our use of this idea is howevermore subtle than the typical applications of probabilistictree embeddings. Instead of moving to a distributionover tree metrics up front, in each phaseℓ, we use the results of [9, 15] todeterministicallyobtain a treeTℓwith edge weightsdTℓ(e), such that the resulting tree metric dominatesd and

∑

e=(i,i′) dTℓ(i, i′)ze,tℓ =

O(log n)∑

e deze,tℓ . As we show in Section 4, this deterministic choice allows toextend our algorithmand analysis effortlessly to the setting where the latency-cost in the objective function is measured by amore general function (e.g., theLp-norm) of the client-latencies. The algorithm is describedin detail asAlgorithm 1, and utilizes the following results. Letτmax = maxj τj.

Theorem 2.1 ([9, 15]) Given any edge weightszee∈E , one can deterministically construct a weightedtreeT having leaf-setF ∪ r, leading to a tree metric,dT (·), such that, for anyi, i′ ∈ F ∪ r, we have:(i) dT (i, i′) ≥ di,i′ , and (ii)

∑

e=(i,i′)∈E dT (i, i′)zi,i′ = O(log n)

∑

e deze.

Theorem 2.2 ([17]) Consider a treeT rooted atr with n leaves, subsetsG1, . . . , Gp of leaves, and frac-tional valuesze on the edges ofT satisfyingz(δ(S)) ≥ νj for every groupGj and node-setS such thatGj ⊆ S, whereνj ∈

[

12 , 1

]

. There exists a randomized polytime algorithm, henceforthcalled the GKRalgorithm, that returns a rooted subtreeT ′′ ⊆ T such that (i)Pr[e ∈ T ′′] ≤ ze for every edgee ∈ T ; and(ii) Pr[T ′′ ∩Gj = ∅] ≤ exp

(

−νj

64 log2 n

)

for every groupGj .

Algorithm 1 Given: a fractional solution(x, y, z) to (P) (withC∗j , L

∗j , Nj , andτj defined as above for each clientj).

A1. In each phaseℓ = 0, 1, . . . ,N := ⌈log2(2τmax) + 4 log2 m⌉, we do the following. Lettℓ = min2ℓ,T.

A1.1. Use Theorem 2.1 with edge weightsze,tℓ to obtain a treeTℓ =(

V (Tℓ), E(Tℓ))

. ExtendTℓ to a treeT ′ℓ by

adding a dummy leaf edge(i, vi) of costfi to Tℓ for each facilityi. LetE′ = (i, vi) : i ∈ F.

A1.2. Map the LP-assignmentze,tℓe∈E to an assignmentz on the edges ofT ′ℓ by setting ze =

∑

e lies on the uniquei-i′ path inTℓzii′,tℓ for all e ∈ E(Tℓ), andze =

∑

t≤tℓyi,t for all e = (i, vi) ∈ E′. Note

that∑

e∈E(Tℓ)dTℓ

(e)ze =∑

e=(i,i′) dTℓ(i, i′)ze,tℓ = O(log n)

∑

e deze,tℓ = O(log n)tℓ.

A1.3. DefineDℓ = j : τj ≤ tℓ. For each clientj ∈ Dℓ, we define the groupN ′j = vi : i ∈ Nj. We

now compute a subtreeT ′ℓ of T ′

ℓ as follows. We obtainN := log2 m subtreesT ′′1 , . . . , T

′′N . Each treeT ′′

r

is obtained by executing the GKR algorithm192 log2 n times on the treeT ′ℓ with groupsN ′

jj∈Dℓ, and

taking the union of all the subtrees returned. Note that we may assume thati ∈ T ′′r iff (i, vi) ∈ T ′′

r . SetT ′ℓ to be any tree inT ′′

1 , . . . , T′′N satisfying (i)

∑

(i,vi)∈E(T ′

ℓ) fi ≤ 40 · 192 log2 n

∑

(i,vi)∈E′ fizi,vi and (ii)∑

e∈E(T ′

ℓ)\E′ dTℓ

(e) ≤ 40 · 192 log2 n∑

e∈E(Tℓ)dTℓ

(e)ze; if no such tree exists, the algorithm fails.

A1.4. Now remove all the dummy edges fromT ′ℓ , open all the facilities in the resulting tree, and convert the

resulting tree into a tourTourℓ traversing all the opened facilies. For every unconnected client j, we connectj to a facility inNj if some such facility is open (and hence part ofTourℓ).

A2. Return the concatenation of the toursTourℓ for ℓ = 0, 1, . . . ,N shortcutting whenever possible. This induces anordering of the open facilities. If some client is left unconnected, we say that the algorithm has failed.

Analysis. The algorithm may fail in steps A.1.3 and A2. Lemmas 2.4 and 2.5 bound the failure probabilityin each case by1/ poly(m). To bound the expected cost conditioned on success, it suffices to bound theexpectation of the random variable that equals the cost incurred if the algorithm succeeds, and is 0 otherwise.

6

Since each clientj is connected to a facility inNj, the total connection cost is at most4∑

j C∗j . To

bound the remaining components of the cost, we first show thatin any phaseℓ, the z-assignment definedabove “covers” each group inN ′

jj∈Dℓto an extent of at least23 (Claim 2.3). Next, we show in Lemma 2.4

that for every clientj ∈ Dℓ, the probability that a facility inNj is included in the treeT ′ℓ, and hence opened

in phaseℓ, is at least56 ·23 = 5

9 . The facility-cost incurred in a phase isO(log n)∑

i,t fiyi,t, and sinceτmax ≤ T = poly(m), the number of phases isO(logm), so this bounds the facility-opening cost incurred.Also, since the probability thatj is not connected (to a facility inNj) in phaseℓ decreases geometrically(at a rate less than1/2) with ℓ when tℓ ≥ τj, one can argue that (a) with very high probability (i.e.,1− 1/ poly(m)), each clientj is connected to some facility inNj, and (b) the expected latency-cost ofj isat mostO(log n)

∑

e∈E(Tℓ)dTℓ(e)ze = O(log2 n)τj .

Claim 2.3 Consider any phaseℓ. For any subsetS of nodes of the corresponding treeT ′ℓ with r /∈ S, and

anyN ′j ⊆ S wherej ∈ Dℓ, we havez(δ(S)) ≥

∑

i∈Nj ,t≤tℓxij,t ≥ 2/3 (whereδ(S) denotesδT ′

ℓ(S)).

Proof : Let R = S ∩ V (Tℓ), andY ⊆ F be the set of leaves inR. Thenδ(S) = δTℓ(R) ∪(

δ(S) ∩ E′)

.Let δE(Y ) = (i, i′) ∈ E : |i, i′| ∩ Y | = 1. Observe thatz(δTℓ(R)) ≥

∑

e∈δE(Y ) ze,tℓ . This is simplybecause if we sendzii′,tℓ flow along the unique(i, i′) path inTℓ for every(i, i′) ∈ δE(Y ), then we obtaina flow betweenY andF ∪ r \ Y respecting the capacitieszee∈E(Tℓ), and of value equal to the RHSabove. Thus, the inequality follows because the capacity ofany cut containingY must be at least the valueof the flow. Since(x, y, z) satisfies (3), we further have that

∑

e∈δE(Y ) ze,tℓ ≥∑

i∈Y,t≤tℓxij,t. So

z(δ(S)) ≥∑

e∈δTℓ (R)

ze,tℓ +∑

(i,vi)∈δ(S):i/∈Y

(

∑

t≤tℓ

yi,t)

≥∑

i∈Y,t≤tℓ

xij,t +∑

i∈Nj\Y,t≤tℓ

xij,t ≥∑

i∈Nj ,t≤tℓ

xij,t ≥2

3.

Lemma 2.4 In any phaseℓ, with probability 1 − 1/ poly(m), we obtain the desired treeT ′ℓ in step A1.3.

Moreover,Pr[T ′ℓ ∩N ′

j 6= ∅] ≥ 5/9 for all j ∈ Dℓ.

Proof : Consider any treeT ′′r obtained by executing the GKR algorithm192 log2 n times and taking the

union of the resulting subtrees. For brevity, we denote∑

(i,vi)∈E(T ′′r ) fi byF (T ′′

r ), and∑

e∈E(T ′′r )\E′ dTℓ(e)

by dℓ(T′′r ).

For j ∈ Dℓ, let Erj denote the event thatT ′′

r ∩ N ′j is non-empty. By Theorem 2.2, we havePr[Er

j ] ≥

1 − exp(

−νj

64 log2 n· 192 log n

)

≥ 1 − e−3νj ≥ (1 − e−3)νj ≥ 11/18. We also have thatE[

F (T ′′r )

]

≤

192 log2 n∑

(i,vi)∈E′ fizi,vi andE[

dℓ(T′′r )

]

≤ 192 log2 n∑

e∈E(Tℓ)dTℓ(e)ze. LetFr andDr denote respec-

tively the events thatF (T ′′r ) ≤ 40·192 log2 n

∑

(i,vi)∈E′ fizi,vi , anddℓ(T ′′r ) ≤ 40·192 log2 n

∑

e∈E(Tℓ)dTℓ(e)ze.

By Markov’s inequality each event happens with probabilityat least39/40. Thus, for anyj ∈ Dℓ, we getthat

Pr[Erj |(F

r ∧Dr)] ≥ Pr[Erj ∧ Fr ∧Dr] ≥ 1− (7/18 + 1/40 + 1/40) > 5/9. (4)

Now the probability that(Fr ∧Dr)c happens for allr = 1, . . . , N is at most(2/40)N ≤ 1/m4. Hence,with high probability, there is some treeT ′

ℓ := Tr such that bothFr andDr hold, and (4) shows thatPr[T ′

ℓ ∩N ′j 6= ∅] ≥ 5/9 for all j ∈ Dℓ.

Lemma 2.5 The probability that a clientj is not connected by the algorithm is at most1/m4. LetLj bethe random variable equal toj’s latency-cost if the algorithm succeeds and0 otherwise. ThenE

[

Lj

]

=O(log2 n)tℓj , whereℓj (= ⌈log2 τj⌉) is the smallestℓ such thattℓ ≥ τj.

7

Proof : Let Pj be the random variable denoting the phase in whichj gets connected; letPj := N + 1 if

j remains unconnected. We havePr[Pj ≥ ℓ] ≤(

49

)(ℓ−ℓj) for ℓ ≥ ℓj The algorithm proceeds for at least4 log2 m phases after phaseℓj , soPr[j is not connected afterN phases] ≤ 1/m4. Now,

Lj ≤∑

ℓ≤Pj

d(Tourℓ) ≤ 2∑

ℓ≤Pj

∑

e∈E(T ′ℓ)\E

′

dTℓ(e) = O(log n)∑

ℓ≤Pj

∑

e∈E(Tℓ)

dTℓ(e)ze = O(log2 n)∑

ℓ≤Pj

tℓ

so E[

Lj

]

= O(log2 n)

N∑

ℓ=0

Pr[Pj ≥ ℓ] · tℓ ≤ O(log2 n)

[ ℓj∑

ℓ=0

tℓ +∑

ℓ>ℓj

tℓ ·

(

4

9

)(ℓ−ℓj)]

= O(log2 n)tℓj .

Theorem 2.6 Algorithm 1 succeeds with probability1−1/ poly(m), and returns a solution of expected costO(

log n ·maxlog n, logm)

·OPT .

Proof : Lemmas 2.4 and 2.5 show that the failure probability is1/ poly(m). LetY denote the cost incurredif the algorithm succeeds, and 0 otherwise. Sincetℓj ≤ 2τj = O(L∗

j ) for eachj, we haveE[

Y]

=

O(

log n ·maxlog n, logm)

·OPT by Lemma 2.5 and the preceding arguments.

Removing the assumption T = poly(m). We first argue that although (P) has a pseudopolynomialnumber of variables, one can compute a near-optimal solution to it in polynomial time (Lemma 2.7) by con-sidering only (integer) time-values that are powers of(1 + ǫ) (rouhgly speaking). Givenǫ > 0, defineTr =⌈(1 + ǫ)r⌉, and letTS := T0,T1, . . . ,Tkwherek is the smallest integer such thatTk ≥ minn,mdmax.DefineT−1 = 0. Let (P)TS denote (P) whent ranges overTS.

Lemma 2.7 For anyǫ > 0, we can obtain a solution to(P)of cost at most(1+ǫ)OPT in timepoly(input size, 1/ǫ).

Proof : We prove that the optimal value of (P)TS is at most(1 + ǫ)OPT . Since the size of (P)TS ispoly(input size, 1/ǫ) this proves the lemma.

We transform(x, y, z), an optimal solution to (P) to a feasible solution(x′, y′, z′) to (P)TS of cost atmost(1+ǫ)OPT . (In fact, the facility-opening and connection-costs remain unchanged, and the latency-costblows up by a(1+ǫ)-factor.)z′ is simply a restriction ofz to the times inTS, that is,z′e,t = ze,t for eache, t ∈TS. Setx′ij,1 = xij,1, y

′i,1 = yi,1 for all i andj. For eachℓ = 1, . . . , k, facility i, client j, we setx′ij,Tℓ

=∑

Tℓt=Tℓ−1+1 xij,t andy′i,Tℓ

=∑

Tℓt=Tℓ−1+1 yi,t. It is clear that

∑

t∈TS x′ij,t =

∑

t xij,t and∑

t∈TS y′i,t =

∑

t yi,t for all i andj, and moreover for anyTℓ ∈ TS, we have∑

t∈TS:t≤Tℓx′ij,t =

∑

t≤Tℓxij,t. It fol-

lows that(x′, y′, z′) is a feasible solution to (P)TS and∑

i,t∈TS fiy′i,t =

∑

i,t fiyi,t,∑

j,i,t∈TS cijx′ij,t =

∑

j,i,t cijxij,t. To bound the latency cost, note that for anyt > Tℓ−1, we haveTℓ ≤ (1 + ǫ)t, sofor any clientj and facility i,

∑

t∈TS tx′ij,t ≤ xij,1 + (1 + ǫ)

∑

t>1 txij,t ≤ (1 + ǫ)∑

t txij,t. Thus,∑

j,i,t∈TS tx′ij,t ≤ (1 + ǫ)

∑

j,i,t txij,t.

Let (x′, y′, z′) denote an optimal solution to (P)TS. The only changes to Algorithm 1 are in the definitionof the timetℓ and the number of phasesN . (Of course we now work with(x′, y′, z′), andC∗

j , L∗j are

defined in terms of(x′, y′, z′) now.) The idea is to definetℓ so that one can “reach” theτj of every clientjin O(logm) phases; thus, one can terminate inO(logm) phases and thereby obtain the same approximationon the facility-opening cost. LetL = (

∑

j L∗j)/m = (

∑

j,i,t∈TS tx′ij,t)/m. Forx ≤ Tk, defineTS(x) to be

the earliest time inTS that is at leastx; if x ≥ Tk, defineTS(x) := Tk. Note thatTS(x) ≤ (1 + ǫ)x for allx ≥ 0. We now definetℓ = TS(L ·2ℓ), and set the number of phases toN :=

⌈

log2(2τmax/L) + 4 log2 m⌉

.Note that sinceτj = O(L∗

j), we haveN = O(logm).

Theorem 2.8 For anyǫ > 0, Algorithm 1 with the above modifications succeeds with highprobability andreturns a solution of expected costO

(

log nmaxlog n, logm)

(1 + ǫ)OPT .

8

Proof : The analysis of the facility-opening and connection-cost is exactly as before (sinceN = O(logm)).Defineℓj as the smallestℓ such thattℓ ≥ τj. Note thatℓj ≤

⌊

log2(2τj/L)⌋

(this holds even whenτj ≤L). Hence, the probability thatj is not connected afterN phases is at most1/m4. So as before, thefailure probability is at most1/ poly(m). We have

∑

ℓ≤ℓjtℓ ≤ (1 + ǫ)L

∑

ℓ≤ℓj2ℓ ≤ 2(1 + ǫ)tℓj , and

∑

ℓ>ℓjtℓ(

49

)(ℓ−ℓj) = O(tℓj). Thus, the inequalities involvingLj andE[

Lj

]

in Lemma 2.5 are still valid,

and we obtain the same bound onE[

Lj

]

as in Lemma 2.5. Note thattℓj ≤ 2(1 + ǫ)τj = O(L∗j ) when

τj ≥ L. Thus,∑

j E[

Lj

]

≤ O(log2 n)[

m · L+∑

j:τj>t0O(L∗

j )]

= O(log2 n)L∗.

Inappproximability of MLUFL. We argue that any improvement in the guarantee obtained in Theorem 2.6would yield an improvement in the approximation factor forGST. We reduceGST to MGL, the special caseof MLUFL mentioned in Section 1, where we have groupsGj ⊆ F and the goal is to order the the facilitiesso as to minimize the sum of the covering times of the groups. (Note that Theorem 2.6 implies anO(log2 n)-approximation forMGL.) Recall that we may assume that the groups inMGL are disjoint, in which case theconnection costs form a metric.

Theorem 2.9 Given aρn,m-approximation algorithm forMGL with (at most)n nodes andm groups, wecan obtain anO(ρn,m logm)-approximation algorithm forGST with n nodes andm groups. Thus, thepolylogarithmic inapproximability ofGST [22] implies thatMGL, and henceMLUFL even with metric con-nection costs, cannot be approximated to a factor better than Ω(logm), even when the time-metric arisesfrom a hierarchically well-separated tree, unlessNP⊆ ZTIME (npolylog(n)).

Our proof of the above theorem is LP-based and is deferred to Appendix A. Gupta et al. [21] indepen-dently arrived at the above theorem via a combinatorial proof.

2.2 MLUFL with related metrics

Here, we consider theMLUFL problem when the facilities, clients, and the rootr are located in a commonmetric space that defines the connection-cost metric (onF ∪ D ∪ r), and we haveduv = cuv/M for allu, v ∈ F∪D∪r. We call this problem,relatedMLUFL, and design anO(1)-approximation algorithm for it.

The algorithm follows a similar outline as Algorithm 1. As before, we build the tour on the openfacilities by concatenating tours obtained by “Eulerifying” trees rooted atr of geometrically increasinglength. At a high level, the improvement in the approximation arises because one can now obtain thesetrees without resorting to Theorems 2.1 and 2.2 and losingO(log n)-factors in process. Instead, sincethe d- and c- metrics are related, we can obtain a group Steiner tree on the relevant groups by using aSteiner tree algorithm (in a manner similar to the LP-rounding algorithms in [27, 20]). We now defineNj = i :

∑

t xij,t > 0, cij ≤ 3C∗j , andτj = 6L∗

j . Ideally, in each phaseℓ, we want to connect theNj

groups for allj such thatτj ≤ tℓ := 2ℓ. But to obtain a low-cost solution, we do a facility-location-styleclustering of the set of clients withτj ≤ tℓ and build a treeTℓ that connects theNjs of only the clustercenters: we contract theseNjs and build an MST (in thed-metric) on them, and then connect eachNj

internally (in thed-metric) using intracluster edges incident onj. Here we crucially exploit the fact that thed- andc-metrics are related.

Deciding which facilities to open is tricky because groupsNj andNk created in different phases couldoverlap, and we could haveC∗

j ≪ C∗k but τj ≫ τk; so if we openi ∈ Nj and use this to also servek, then

we must connecti to someTℓ (without increasing itsd-cost by much) wheretℓ = O(τk). We consider thecollectionC of cluster centers created in all the phases, and pick a maximal subsetC′ ⊆ C that yields disjointclusters by greedily considering clusters in increasingC∗

j order. We open the cheapest facility inNj for allj ∈ C′, and attach it to the treeTℓ, whereℓ is theearliestphase such that there is some clusterNk created

9

in that phase that was removed (fromC) whenNj was included inC′ (becauseNk ∩Nj 6= ∅). Sinced andc are related, one can bound the resulting increase in thed-cost ofTℓ. Finally, we convert these augmentedTℓ-trees to tours and concatenate these tours.

We now describe the algorithm in detail. We have not sought tooptimize the approximation ratio. Whenwe refer to an edge or a node below, we mean an edge or node of thecomplete graph onF ∪D∪r. RecallthatNj = i :

∑

t xij,t > 0, cij ≤ 3C∗j , andτj = 6L∗

j . So∑

i∈Nj

∑

t xij,t ≥23 , and

∑

i∈Nj ,t≤τjxij,t ≥

12 .

R1. For each timetℓ = 2ℓ, whereℓ = 0, 1, . . . , ⌈log T⌉, we do the following. DefineDℓ = j : τj ≤tℓ \

(⋃

0≤ℓ′<ℓ Cℓ′)

(where the union of an empty collection is∅).

• (Clustering) We cluster the facilities in⋃

j∈DℓNj as follows. We pickj ∈ Dℓ with smallestC∗

j valueand form a cluster aroundj consisting of the facilities inNj . For every clientk ∈ Dℓ (including j)such thatcjk ≤ 30C∗

k (note thatC∗k ≥ C∗

j ), we removek from Dℓ, setσ(k) = j, and recurse onthe remaining clients inDℓ until no client is left inDℓ. Let Cℓ denote the set of cluster centers (i.e.,j ∈ Dℓ : σ(j) = j). Note that for two clientsj andj′ in Cℓ, Nj ∩Nj′ is ∅.

• (Building a group Steiner tree Tℓ on Njj∈Cℓ ) We contract the clustersNj for j ∈ Cℓ into supern-odes, and build a minimum spanning tree (MST)T ′′

ℓ connectingr and these supernodes. Next, weuncontract the supernodes, and for eachj ∈ Cℓ, we add edges joiningj to every facilityi ∈ Nj thathas an edge incident to it inT ′′

ℓ . This yields the treeTℓ.

R2. (Opening facilities) Let C =⋃

ℓ Cℓ. (Note that a client appears in at most one of theCℓ sets.) Wecannot open a facility in every cluster centered around a client inC, since forj ∈ Cℓ andk ∈ Cℓ′ , Nj

andNk need not be disjoint. So we select a subsetC′ ⊆ C such that for any twoj, k in C′, the setsNj

andNk are disjoint. This is done as follows. We initializeC′ ← ∅. Pick the clientj ∈ C with smallestC∗j value and add it toC′. We delete fromC every clientk ∈ C (including j) such thatNk ∩ Nj 6= ∅,

settingnbr(k) = j, and recurse on the remaining set of clients until no client is left inC.

Consider eachj ∈ C′. We open the facilityi ∈ Nj with smallestfi. Let ℓ be the smallest index suchthat there is somek ∈ Cℓ with nbr(k) = j. We connecti to Tℓ by adding thefacility edge(i, k) to it.Let T ′

ℓ denoteTℓ augmented by all such facility edges.

R3. We obtain a tour connecting all the open facilities, by converting each treeT ′ℓ into a tour, and concate-

nating the tours forℓ = 0, . . . , ⌈T⌉ (in that order).

R4. For every clientj ∈ C, we assignj to the facility opened fromNnbr(j). For every clientj /∈ C, weassignj to the same facility asσ(j).

Theorem 2.10 For relatedMLUFL, one can round(x, y, z) to get a solution with facility-opening cost atmost32

∑

i,t fiyi,t, where each clientj incurs connection-cost at most39C∗j and latency-cost at most64τj =

384L∗j . Thus, we obtain anO(1)-approximation algorithm for relatedMLUFL.

Proof : The clustersNj for clientsj ∈ C′ are disjoint; each such cluster has facility weight∑

i∈Nj

∑

t yi,t ≥23 , and we open the cheapest facility in the cluster.

Consider a clientj, and let it be assigned to facilityi. If j ∈ C′, theni ∈ Nj , and we havecij ≤ 3C∗j .

If j ∈ C \ C′ with nbr(j) = k, theni ∈ Nk and there is some facilityi′ ∈ Nj ∩ Nk. So cij ≤ cik +ci′k + ci′j ≤ 2 · 3C∗

k + 3C∗j ≤ 9C∗

j . Finally, if j /∈ C andj′ = σ(j), thenσ(j′) is also assigned toi, socij ≤ cij′ + cjj′ ≤ 9C∗

j′ + 30C∗j ≤ 39C∗

j .We next boundd(Tℓ) andd(T ′

ℓ ) for any phaseℓ. For any clientj ∈ Dℓ and any node-setS ⊇ Nj, r /∈ S,we have

∑

e∈δ(S) ze,tℓ ≥∑

i∈S,t≤tℓxij,t ≥

∑

i∈Nj ,t≤τjxij,t ≥

12 . Therefore,(2ze,tℓ) forms a fractional

10

Steiner tree of cost at most2tℓ on the supernodes andr, and hence,d(T ′′ℓ ) ≤ 4tℓ since it is well known that

the cost of an MST is at most twice the cost of a fractional solution to the Steiner-tree LP. Forj ∈ Cℓ, letdegj denote the degree of the clusterNj in T ′′

ℓ . Observe that ife is an edge ofT ′′ℓ joining Nj andNk (so

j, k ∈ Cℓ), thence ≥ 24maxC∗j , C

∗k, since30maxC∗

j , C∗k ≤ cjk ≤ 3C∗

j + ce + 3C∗k . So the (d-) cost

of adding the additional edges toT ′′ℓ is at most 1M ·

∑

j∈Cℓdegj ·3C

∗j ≤ d(T ′′

ℓ )/4, and hence,d(Tℓ) ≤ 5tℓ.Now consider the cost of adding facility edges toTℓ in step R2. For each facility edge(i, k) added, we

know thati ∈ Nnbr(k), k ∈ Cℓ, andk is assigned toi. So we havecik ≤ 9C∗k . Observe that each client

k ∈ Cℓ is responsible for at most one such facility edge. So thed-cost of these facility edges is at most1M ·

∑

j∈Cℓ9C∗

j ≤1M

∑

j∈Cℓdegj ·9C

∗j ≤

34 · d(T

′′ℓ ). Thus,d(T ′

ℓ ) ≤ d(Tℓ) +34 · d(T

′′ℓ ) ≤ 8tℓ.

Finally, we prove that the latency cost of any clientj is at most64τj = 384L∗j . Let j be assigned

to facility i. Let ℓ be the smallest index such thatj ∈ Dℓ, so tℓ ≤ 2τj. We first argue that ifi is partof the treeT ′

ℓ′ , thenℓ′ ≤ ℓ. If j ∈ C, this follows since we know thati ∈ Nnbr(j) and ℓ′ = minr :∃k ∈ Cr with nbr(k) = nbr(j). If j /∈ C, then we know thatσ(j) ∈ Cℓ is also assigned toi, andso by the preceding argument, we again have thatℓ′ ≤ ℓ. Thus, the latency-cost ofj is bounded by∑ℓ′

r=0 2d(T′r ) ≤ 16

∑ℓ′

r=0 tr ≤ 32tℓ′ ≤ 64τj .

2.3 MLUFL with a uniform time-metric

We now consider the special case ofMLUFL, referred to asuniform MLUFL, where the time-metricd isuniform, that is,dii′ = 1 for all i, i′ ∈ F ∪ r. When the connection costs form a metric, we call it themetric uniformMLUFL. We consider the following simpler LP-relaxation of the problem, where the timetnow ranges from1 to n.

min∑

i,t

fiyi,t +∑

j,i,t

(cij + t)xij,t subject to (Unif-P)

∑

i,t

xij,t ≥ 1 ∀j; xij,t ≤ yi,t ∀i, j, t;∑

i

yi,t ≤ 1 ∀t; xij,t, yi,t ≥ 0 ∀i, j, t.

Let (x, y) be an optimal solution to (Unif-P), andOPT be its value. LetC∗j =

∑

i,t cijxij,t, L∗j =

∑

i,t txij,t. As stated in the introduction, uniformMLUFL generalizes: (i) set cover, when the facility andconnection costs are arbitrary; (ii)MSSC, when the facility costs are zero (ZFC MLUFL); and (iii) metricUFL, when the connection costs form a metric. We obtain approximation bounds for uniformMLUFL, ZFCMLUFL, and metricMLUFL (Theorems 2.11 and 2.14) that complement these observations.

The main result of this section is Theorem 2.12, which shows that aρUFL-approximation algorithm forUFL and aγ-approximation algorithm forZFC MLUFL (with metric connection costs) can be combined toyield a(ρUFL +2γ)-approximation algorithm for metric uniformMLUFL. TakingρUFL = 1.5 [8] andγ = 9(part (ii) of Theorem 2.11), we obtain a 19.5-approximationalgorithm. We improve this to10.773 by usinga more refined version of Theorem 2.12, which capitalizes on the asymmetric approximation bounds thatone can obtain for different portions of the total cost inUFL andZFC MLUFL.

We note that by considering each(i, t) as a facility, since the connection costscij + t form a metric, onecan view metricMLUFL as a variant of metricUFL, and use the ideas in [3] to devise anO(1)-approximationfor this variant. We instead present our alternate algorithm based on the reduction in Theorem 2.12, becausethis reduction is quite robust and versatile. In particular, it allows us to: (a) handle certain extensions ofthe problem, e.g., the setting where we have non-uniform latency costs (see Section 4), for which the abovereduction fails since we do not necessarily obtain metric connection costs, and (b) devise algorithms for theuniform latency versions of other facility location problems, where the cost of a facility does not dependon the client-set assigned to it (so one can assign a client toany open facility freely without affecting thefacility costs). For instance, consider uniformMLUFL with the restriction that at mostk facilities may be

11

opened: our technique yields a(ρkMed + 2γ) approximation for this problem, using aρkMed-approximationfor k-median.

Theorem 2.11 One can obtain:(i) anO(lnm)-approximation algorithm for uniformMLUFL with arbitrary facility- and connection- costs.(ii) a solution of cost at most 11−α

∑

j C∗j + 4

α

⌈

1α

⌉∑

j L∗j for ZFCMLUFL, for any parameterα ∈ (0, 1).

Thus, settingα = 89 , yields solution of cost at most9 ·OPT .

We defer the proof of Theorem 2.11 to the end of the section, and focus first on detailing the aforemen-tioned reduction.

Theorem 2.12 Given aρUFL-approximation algorithmA1 for UFL, and aγ-approximation algorithmA2

for uniformZFC MLUFL, one can obtain a(ρUFL+2γ)-approximation algorithm for metric uniformMLUFL.

Proof : Let I denote the metric uniformMLUFL instance, andO∗ denote the cost of an optimal integersolution. LetIUFL be theUFL instance obtained formI by ignoring the latency costs, andIZFC be theZFCMLUFL instance obtained fromI by setting all facility costs to zero. LetO∗

UFLandO∗

ZFCdenote respectively

the cost of the optimal (integer) solutions to these two instances. Clearly, we haveO∗UFL

, O∗ZFC≤ O∗. We

useA1 to obtain a near-optimal solution toIUFL: let F1 be the set of facilities opened and letσ1(j) denotethe facility inF1 to which clientj is assigned. So we have

∑

i∈F1fi+

∑

j cσ1(j)j ≤ ρUFL ·O∗UFL

. We useA2

to obtain a near-optimal solution toIZFC: let F2 be the set of open facilities,σ2(j) be the facility to whichclient j is assigned, andπ(i) be the position of facilityi. So we have

∑

j

(

cσ2(j)j + π(σ2(j)))

≤ γ ·O∗ZFC

.We now combine these solutions as follows. For each facilityi ∈ F2, let µ(i) ∈ F1 denote the facility

in F1 that is nearest toi. We open the setF = µ(i) : i ∈ F2 of facilities. The position of facilityi ∈ F is set tomini′∈F2:π(i′)=i π(i

′). Each facility inF is assigned a distinct position this way, but somepositions may be vacant. Clearly we can always convert the above into a proper ordering ofF where eachfacility i ∈ F occurs at positionκ(i) ≤ mini′∈F2:π(i′)=i π(i

′). Finally, we assign each clientj to thefacility φ(j) = µ(σ2(j)) ∈ F . Note thatκ(φ(j)) ≤ π(σ2(j)) (by definition). For a clientj, we now havecφ(j)j ≤ cσ2(j)µ(σ2(j)) + cσ2(j)j ≤ cσ2(j)σ1(j) + cσ2(j)j ≤ cσ1(j)j + 2cσ2(j)j. Thus, the total cost of theresulting solution is at most

∑

i∈F1fi +

∑

j

(

cσ1(j)j + 2cσ2(j)j + π(σ2(j)))

≤ (ρUFL + 2γ) · O∗.

We call an algorithm a(ρf , ρc)-approximation algorithm forUFL if given an LP-solution toUFL withfacility- and connection- costsF ∗ andC∗ respectively, it returns a solution of cost at mostρfF

∗ + ρcC∗.

Similarly, we say that an algorithm is a(γc, γl)-approximation algorithm for uniformZFC MLUFL if givena solution to (P) with connection- and latency- costsC∗ andL∗ respectively, it returns a solution of cost atmostγcC∗ + γlL

∗. The proof of Theorem 2.12 easily yields the following more general result.

Corollary 2.13 One can combine a(ρf , ρc)-approximation algorithm forUFL, and a(γc, γl)-approximationalgorithm for uniform ZFCMLUFL, to obtain a solution of cost at mostmaxρf , ρc + 2γc, γl ·OPT .

Proof : The proof mimics the proof of Theorem 2.12. The only new observation is that(x, y) yields anLP-solution to (i)IUFL with facility cost

∑

i,t fiyi,t and connection cost∑

j,i,t cijxij,t; and (ii) IZFC withconnection cost

∑

j,i,t cijxij,t and latency cost∑

j,i,t txij,t. Thus, applying the construction in the proof ofTheorem 2.12 yields a solution of total cost at most

ρf∑

i,t

fiyi,t + (ρc + 2γc)∑

j,i,t

cijxij,t + γl∑

j,i,t

txij,t ≤ maxρf , ρc + 2γc, γl ·OPT .

Combining the( ln(1/β)

1−β , 31−β

)

-approximation algorithm forUFL [29] with the(

11−α ,

4α

⌈

1α

⌉)

-approximationalgorithm forZFC MLUFL (part (ii) of Theorem 2.11) gives the following result.

12

Theorem 2.14 For anyα, β ∈ (0, 1), one can obtain a solution of costmax ln(1/β)

1−β , 31−β + 2

1−α ,4α

⌈

1α

⌉

·OPT . Thus, takingα = 0.7426, β = 0.000021, we obtain a10.773-approximation algorithm.

2.3.1 Proof of Theorem 2.11

The following lemma will often come in handy.

Lemma 2.15 Let (x, y) be a solution satisfying∑

i yi,t ≤ k for every timet, wherek ≥ 0 is an integer, andall the other constraints of(Unif-P). Then, one can obtain a feasible solution(x′, y′) to (Unif-P) such that(i)

∑

i,t fiy′i,t =

∑

i,t fiyi,t; (ii)∑

j,i,t cijx′ij,t =

∑

j,i,t cij xij,t; and (iii)∑

j,i,t tx′ij,t ≤ k ·

∑

j,i,t txij,t.

Proof : For each timet, defineSt = (i, t) : yi,t > 0. The idea is to simply “spread out” theSt sets. LetS0 =

⋃

t St be an ordered list where all the(·, t) pairs are listed before any(·, t + 1) pair, and the pairs for

a givent (i.e., (i, t) ∈ St) are listed in arbitrary order. LetT0 =⌈

∑

i,t yi,t

⌉

. We divide the pairs inS0 into

T0 groups as follows. Initializeℓ ← 1, S ← S0. For a setA of (i, t) pairs, we define they-weight ofA asy(A) =

∑

(i,t)∈A yi,t, and thexj-weight ofA as∑

(i,t)∈A xij,t. If 0 < y(S) ≤ 1, then we setGℓ = S toend the grouping process. Otherwise, groupGℓ includes all pairs ofS, taken in order starting from the firstpair, stopping when the totaly-weight of the included pairs becomes at least 1; all the included pairs are alsodeleted fromS. If the y-weight ofGℓ now exceeds 1, then we split the last pair(i, t) into two copies: weinclude the first copy inGℓ and retain the second copy inS, and distributeyi,t across they-weight of the twocopies so thaty(Gℓ) is nowexactly1. (Thus, the newy-weight ofS is precisely its oldy-weight−1.) Also,for each clientj, we distributexij,t across thexj-weight of the two copies arbitrarily while maintaining thatthe xj-weight of each copy is at most itsy-weight. We updateℓ← ℓ+ 1, and continue in this fashion withthe current (i.e., ungrouped) list of pairsS. Note that an(i, t) pair inS0 may be split into at most two copiesabove (that lie in consecutive groups). To avoid notationalclutter, we call both these copies(i, t) and useyℓi,t to denote they-weight of the copy in groupGℓ (which is equal to the originalyi,t if (i, t) is not split).Analogously, we usexℓij,t to denote thexj-weight of the copy of(i, t) in groupGℓ.

For every facilityi, clientj, andℓ = 1, . . . , T0, we sety′i,ℓ =∑

t:(i,t)∈Gℓyℓi,t andx′ij,ℓ =

∑

t:(i,t)∈Gℓxℓij,t,

so we havex′ij,ℓ ≤ y′i,ℓ. It is clear that∑

ℓ y′i,ℓ =

∑

t yi,t and∑

ℓ x′ij,ℓ =

∑

t xij,t for every facility i andclient j. Thus,(x′, y′) is feasible to (Unif-P), and parts (i) and (ii) of the lemma hold. To prove part (iii),note that if (some copy of)(i, t) is in Gℓ, thenℓ ≤ kt, since then we have

(⋃ℓ−1

r=1Gr

)

⊂⋃t

t′=1 St′ and soℓ− 1 < kt. Thus, for any clientj, we have

∑

i,ℓ

ℓx′ij,ℓ =∑

i,ℓ

ℓ(

∑

t:(i,t)∈Gℓ

xℓij,t)

=∑

i,t

∑

ℓ:(i,t)∈Gℓ

ℓxℓij,t ≤∑

i,t

kt∑

ℓ:(i,t)∈Gℓ

xℓij,t = k ·∑

i,t

txij,t.

Proof of part (i) of Theorem 2.11 : We round the LP-optimal solution(x, y) by using filtering followed bystandard randomized rounding. Clearly, we may assume that

∑

i′,t xi′j,t = 1 andyi,t = maxj xij,t for everyi, j, t. Also, we may assume that if

∑

i yi,t+1 > 0, then∑

i yi,t = 1, because otherwise for some facilityiand someǫ > 0, we may decreaseyi,t+1 by ǫ and increaseyi,t by ǫ, and modify thexij,t+1, xij,tj valuesappropriately so as to maintain feasibility, without increasing the total cost. DefineNj = (i, t) : cij + t ≤2(C∗

j + L∗j) for a clientj. The algorithm is as follows.

U1. For each(i, t), we setYi,t = 1 independently with probabilitymin4 lnm · yi,t, 1.

U2. Considering each clientj, if (i, t) ∈ Nj : Yi,t = 1 = ∅, then setYij ,1 = 1 whereij is such thatfij = min(i,t)∈Nj

fi (note that(ij , 1) ∈ Nj). LetSj = (i, t) ∈ Nj : Yi,t = 1 (which is non-empty).Assign each clientj to the(i, t) pair inSj with minimumcij + t value, i.e., setXij,t = 1.

13

U3. LetK = maxt∑

i Yi,t. Use Lemma 2.15 to convert(X,Y ) into a feasible integer solution to (Unif-P).

Let Cj andLj denote respectively the connection cost and latency cost ofclient j in (X,Y ). We arguethat (i) E

[∑

i,t fiYi,t

]

= O(lnm) ·∑

i,t fiyi,t, (ii) with probability 1,Cj + Lj ≤ 2(C∗j + L∗

j) for everyclient j, and (iii) K = O(lnm) with high probability, and in expectation. The theorem thenfollows fromLemma 2.15.

For any clientj, we have∑

(i,t)∈Njxij,t ≥

12 (by Markov’s inequality). Thus,fij ≤ 2

∑

(i,t)∈Njfiyi,t

andPr[∑

(i,t)∈NjYi,t = 0] is at moste−4 lnm· 1

2 = 1/m2. The expected cost of opening facilities instep U1 is clearly at most4 lnm ·

∑

i,t fiyi,t. The expected facility-opening cost in step U2 is at mostPr[facility is opened in step U2]·

∑

j fij ≤ m· 1m2 ·2m

∑

i,t fiyi,t. SoE[∑

i,t fiYi,t

]

= O(lnm)·∑

i,t fiyi,t.Since we always open some(i, t) pair inNj , we haveCj + Lj ≤ 2(C∗

j + L∗j ) for every clientj.

Let S = t :∑

i yi,t > 0. Note that|S| ≤ 1 +∑

i,t yi,t ≤ 1 +∑

j,i,t xij,t = m + 1. After step U1,we haveE

[∑

i Yi,t

]

≤ 4 lnm for every timet ∈ S. Since theYi,t random variables are independent, wealso havePr[

∑

i Yi,t > 8 lnm] ≤ 1/m2 for all t ∈ S (and also,E[

maxt∈S∑

i Yi,t

]

= O(lnm)). Thus,after step U2, we havePr[

∑

i Yi,1 > 8 lnm] ≤ 1/m2 + 1/m andPr[∑

i Yi,t > 8 lnm] ≤ 1/m2 for allt ∈ S, t > 1. Hence,Pr[K > 8 lnm] ≤ 2/m. (This also shows thatE

[

K]

= O(lnm).)

Proof of part (ii) of Theorem 2.11 : We round(x, y) by applying filtering [23] followed by Lemma 2.15 toreduce the problem to aMSSC problem, and then use the result of Feige et al. [16] to obtaina near-optimalsolution to thisMSSC problem. Let

min∑

j,t

txj,t s.t.∑

t

xj,t ≥ 1 ∀j, xj,t ≤∑

S:j∈S

yS,t ∀j, t,∑

S

yS,t ≤ 1 ∀t, x, y ≥ 0. (P1)

denote the standard LP-relaxation ofMSSC [16] (herej indexes the elements,S indexes the sets, andtindexes time). Feige et al. showed that given a solution(x, y) to (P1), one can obtain in polytime an integersolution of cost at most4 ·

∑

j,t txj,t.The rounding algorithm forZFC MLUFL is as follows. DefineNj = i : cij ≤ C∗

j /(1 − α), so∑

i∈Nj ,txij,t ≥ α. For all i, j, t, setyi,t = yi,t/α, andxij,t = xij,t/α if i ∈ Nj and xij,t = 0 otherwise.

It is easy to see that(x, y) satisfies∑

i yi,t ≤1α for all t, and all the other constraints of (Unif-P). We use

Lemma 2.15 to convert(x, y) to a feasible solution(x′, y′) to (Unif-P). Next, we extract a solution to (P1)from (x′, y′). We identify facility i with the setj : i ∈ Nj, and setxj,t =

∑

i∈Njx′ij,t. Now (x, y′) is a

feasible solution to (P1). Finally, we round(x, y′) to an integer solution. This yields the orderingy = (yi,t)of the facilities. For each clientj, if j is first covered by seti (soi ∈ Nj) at time (or position)t in theMSSCsolution, then we setxij,t = 1.

Analysis. Since a clientj is always assigned to a facility inNj, the connection cost ofj is bounded byC∗j /(1 − α). To bound the latency cost, first we bound the cost of(x, y). Since we modify the assign-

ment of a clientj by transferring weight from farther facilities to nearer ones, it is clear that∑

i,t cijxij,t ≤∑

i,t cijxij,t. Also, clearly∑

j,i,t txij,t ≤1α ·

∑

j,i,t txij,t. and∑

i yi,t ≤1α . Thus, applying Lemma 2.15

yields (x′, y′) satisfying∑

j,i,t tx′ij,t ≤

⌈

1α

⌉

1α ·

∑

j,i,t txij,t. The result of [16] now implies that∑

j,i,t txij,t ≤ 4∑

j,t txj,t = 4∑

j,i,t tx′ij,t ≤

4α

⌈

1α

⌉∑

j txij,t.

3 LP-relaxations and algorithms for the minimum-latency problem

In this section, we consider the minimum-latency (ML) problem and apply our techniques to obtain LP-based insights and algorithms for this problem. We give two LP-relaxations forML with constant integrality

14

gap. The first LP (LP1) is a specialization of (P) toML, and to bound its integrality gap, we only need thefact that the natural subtour elimination LP forTSP has constant integrality gap. The second LP (LP2P )has exponentially-many variables, one for every path (or tree) of a given length bound, and the separationoracle for the dual problem corresponds to an (path- or tree-) orienteering problem. We prove that even abicriteria approximation for the orienteering problem yields an approximation forML while losing a constantfactor. (The same relationship also holds betweenMGL and “group orienteering”.) As mentioned in theIntroduction, we believe that our results shed new light onML and opens upML to new venues of attack.Moreover, as shown in Section 4 these LP-based techniques can easily be used to handle more generalvariants ofML, e.g.,k-route ML with Lp-norm latency-costs (for which we give the first approximationalgorithm). We believe that our LP-relaxations are in fact (much) better than what we have accounted for,and conjecture that the integrality gap of both (LP1) and (LP2P ) is at most 3.59, which is the currently bestknown approximation factor forML.

LetG = (D∪r, E) be the complete graph onN = |D|+1 nodes with edge weightsde that form ametric. Letr be the root node at which the path visiting the nodes must originate. We usee to indexE andj to index the nodes. In both LPs, we have variablesxj,t for t ≥ djr to denote ifj is visited at timet (wheret ranges from1 to T); for convenience, we think ofxj,t as being defined for allt, with xj,t = 0 if djr > t.(As in Section 2.1, one can move to a polynomial-size LP losing a(1 + ǫ)-factor.)

A compact LP. As before, we use a variableze,t to denote ife has been traversed by timet.

min∑

j,t

txj,t subject to (LP1)

∑

t

xj,t ≥ 1 ∀j;∑

e

deze,t ≤ t ∀t;∑

e∈δ(S)

ze,t ≥∑

t′≤t

xj,t′ ∀t, S ⊆ D, j; x, z ≥ 0.

Theorem 3.1 The integrality gap of(LP1) is at most10.78.

Proof : Let (x, z) be an optimal solution to (LP1), andL∗j =

∑

t txj,t. Forα ∈ [0, 1], define theα-pointof j, τj(α), to be the smallestt such that

∑

t′≤t xjt′ ≥ α. Let Dt(α) = j : τj(α) ≤ t. We round(x, z)as follows. We pickα ∈ (0, 1] according to the density functionq(x) = 2x. At each timet, we utilize the32 -integrality-gap of the subtour-elimination LP for TSP andthe parsimonious property (see [33, 30, 18, 6]),to round2z

α and obtain a tour onr ∪Dt(α) of costCt(α) ≤3α ·

∑

e deze,t ≤3tα . We now use Lemma 3.2

to combine these tours.

Lemma 3.2 ([19] paraphrased) LetTour1, . . . ,Tourk be tours containingr, withTouri having costCi andcontainingNi nodes, whereN0 := 1 ≤ N1 ≤ . . . ≤ Nk = N . One can find a subsetTouri1 , . . . ,Tourib=k

of tours, and a way of concatenating them that gives total latency at most3.592∑

iCi(Ni −Ni−1).

The tours we obtain for the different times are nested (as theDt(α)s are nested). So∑

t≥1 Ct(α)(|Dt(α)|−|Dt−1(α)|) =

∑

j

∑

t:j∈Dt(α)\Dt−1(α)Ct(α) =

∑

j Cτj(α)(α) ≤ 3∑

jτj(α)α . Thus, using Lemma 3.2, and

taking expectation overα (note thatE[ τj(α)

α

]

≤ 2L∗j ), we obtain that the total latency-cost is at most

(3.59 · 3)∑

j L∗j .

Note that in the above proof we did not need any procedure to solve k-MST or its variants, but ratherjust needed the integrality gap for the subtour-elimination LP to be a constant. Also, we can modify therounding procedure to ensure that (latency-cost ofj) ≤ 18τj(0.5) for each clientj, as follows. (Such aguarantee is useful to bound the total cost when its measuredas anLp norm for p > 1; see Section 4.)We now only consider timestℓ = 2ℓ. Recall that for anyα ∈ (0, 1), at each timet, we can obtain a touron r ∪ Dt(α) of costCt(α) ≤

3tα . We take this tour fortℓ, and traverse the resulting tour randomly

clockwise or anticlockwise (this choice can easily be derandomized), and concatenate all these tours. Let

15

ℓj(α) be the smallestℓ such thattℓ ≥ τj(α). So the (expected) latency-cost ofj is at most∑

ℓ<ℓj(α)3tℓα +

12 ·

3tℓj (α)

α ≤4.5tℓj (α)

α ≤9τj(α)

α . Fixingα = 0.5, we obtain a 36-approximation with the per-client guaranteethat (latency-cost ofj) ≤ 18τj(0.5) for eachj.

An exponential-size LP: relating the orienteering and latency problems. LetPt andTt denote respec-tively the collection of all (simple) paths and trees rootedat r of length at mostt. For each pathP ∈ Pt, weintroduce a variablezP,t that indicates ifP is the path used to visit the nodes with latency-cost at mostt.

min∑

j,t

txj,t (LP2P )

s.t.∑

t

xj,t ≥ 1 ∀j

∑

P∈Pt

zP,t ≤ 1 ∀t (5)

∑

P∈Pt:j∈P

zP,t ≥∑

t′≤t

xj,t′ ∀j, t (6)

x, z ≥ 0.

max∑

j

αj −∑

t

βt (LD2)

s.t. αj ≤ t+∑

t′≥t

θj,t′ ∀j, t (7)

∑

j∈P

θj,t ≤ βt ∀t, P ∈ Pt (8)

α, β, θ ≥ 0. (9)

(5) and (6) encode that at most one path may be chosen for any timet, and that every nodej visited at timet′ ≤ t must lie on this path. (LD2) is the dual LP with exponentiallymany constraints. Let(LP2T ) be theanalogue of (LP2P ) with tree variables, where we have variableszQ,t for everyQ ∈ Tt, and we replace alloccurrences ofzP,t in (LP2P ) with zQ,t.

Separating over the constraints (8) involves solving a (rooted) path-orienteering problem: for everyt,given rewardsθj,t, we want to determine if there is a pathP rooted atr of length at mostt that gathersreward more thanβt. A (ρ, γ)-path, tree approximation algorithm for the path-orienteering problem is analgorithm that always returns apath, tree rooted atr of length at mostγ(length bound) that gathers rewardat least(optimum reward)/ρ. Chekuri et al. [11] give a(2+ǫ, 1)-path approximation algorithm, whereas [10]design a(1+ǫ, 1+ǫ)-tree approximation for orienteering (note that weighted orienteering can be reduced tounweighted orienteering with a(1+ǫ)-factor loss). We prove that even a(ρ, γ)-tree approximation algorithmfor orienteering can be used to obtain anO(ργ)-approximation forML. First, we show how to compute anear-optimal LP-solution. Typically, one argues that, scaling the solution computed by the ellipsoid methodrun on the dual with the approximate separation oracle yields a feasible and near-optimal dual solution,and this is then used to obtain a near-optimal primal solution (see, e.g., [24]). However, in our case, wehave negative terms in the dual objective function, which makes our task trickier: if our (unicriteria)ρ-approximate separation oracle determines that(α, β, θ) is feasible, then although(α, ρβ, θ) is feasible to(LD2), one hasno guarantee on the value of this dual solution. Instead, the notion of approximation weobtain for the primal solution computed involves bounded violation of the constraints.

Let(

LP2(a,b)P

)

be (LP2P) where we replacePt by Pbt, and the RHS of (5) is nowa. Let(

LP2(a,b)T

)

be

defined analogously. LetOPTP be the optimal value of (LP2P ) (i.e.,(

LP2(1,1)P

)

). Note thatOPTP is alower bound on the optimum latency.

Lemma 3.3 Given a(ρ, γ)-tree approximation for the orienteering problem, one can compute a feasible

solution(x, z) to(

LP2(ρ,γ)T

)

of cost at mostOPT .

16

Proof : Lemma 3.3 Define

Pfeas(ν; a, b) :=

(α, β, θ) : (7), (9),∑

j∈P

θj,t ≤ βt ∀P ∈ Pbt,∑

j

αj − a∑

t

βt ≥ ν

,

Tfeas(ν; a, b) :=

(α, β, θ) : (7), (9),∑

j∈Q

θj,t ≤ βt ∀Q ∈ Tbt,∑

j

αj − a∑

t

βt ≥ ν

.

Note thatOPTP is the largest value ofν such thatPfeas(ν; 1, 1) is feasible. Givenν, (α, β, θ), if therewas an algorithm to either show(α, β, θ) ∈ Pfeas(ν; 1, 1) or exhibit a separating hyperplane, then usingthe ellipsoid method we could optimally solve forOPTP . However, such a separation oracle would solveorienteering exactly.

We use the(ρ, γ)-tree approximation algorithm to give anapproximateseparation oracle in the followingsense. Givenν, (α, β, θ), we either show(α, ρβ, θ) ∈ Pfeas(ν; 1, 1), or we exhibit a hyperplane separating(α, β, θ) andTfeas(ν; ρ, γ). Note thatTfeas(ν; ρ, γ) ⊆ Pfeas(ν; 1, 1). Thus, for a fixedν, the ellipsoidmethod in polynomial time, either certifies thatTfeas(ν; ρ, γ) is empty or returns a point(α, β, θ) with(α, ρβ, θ) ∈ Pfeas(ν; 1, 1). Let us describe the approximate separation oracle first, and then use the abovefact to prove the lemma. First, check if(

∑

j αj − ρβt ≥ ν), (7), and (9) hold, and if not, we use theappropriate inequality as the separating hyperplane between (α, β, θ) andTfeas(ν; ρ, γ). Next, for eacht,we run the(ρ, γ)-tree approximation on the orienteering problem specified by

(

G, de)

, root r, rewardsθj,t, and budgett. If for somet, we obtain a treeQ ∈ Tγt with reward greater thanβt, then we return∑

j∈Q θj,t ≤ βt as the separating hyperplane. If not, then for all pathsP of length at mostt in G, we have∑

j∈P θj,t ≤ ρβt and thus(α, ρβ, θ) ∈ Pfeas(ν; 1, 1).We find the largestν∗ (via binary search) such that the ellipsoid method run forν∗ with our separation

oracle returns a solution(α∗, β∗, θ∗) with (α∗, ρβ∗, θ∗) ∈ Pfeas(ν∗; 1, 1); hence, we haveν∗ ≤ OPTP (byduality). Now forǫ > 0, the ellipsoid method run forν∗ + ǫ terminates in polynomial time certifying theinfeasibility ofTfeas(ν∗+ ǫ; ρ, γ). That is, it generates a polynomial number of inequalities of the form (7),(9), and(

∑

j∈Q θj,t ≤ βt) whereQ ∈ Tγt, which together with the inequality∑

j αj − ρ∑

t βt ≥ ν∗ + ǫconstitute an infeasible system. Applying Farkas’ lemma, equivalently, we get a polynomial sized solution(x, z) to

(

LP2(ρ,γ)T

)

that has cost at mostν∗ + ǫ. Takingǫ small enough (something like1/ exp(input size)so thatln(1/ǫ) is still polynomially bounded), this also implies that(x, z) has cost at mostν∗ ≤ OPTP .This completes the proof of the lemma.

Theorem 3.4 (i) A feasible solution(x, z) to(

LP2(ρ,γ)T

)

(or the corresponding LP-relaxation forMGL) canbe rounded to obtain a solution of expected cost at mostO(ργ) ·

∑

j,t txj,t; (ii) A feasible solution(x, z) to(

LP2(ρ,γ)P

)

can be rounded to obtain a solution of cost at most(3.59 · 2)ργ∑

j,t txj,t.

Proof : We prove part (i) first. (Note that forML, the analysis leading to Theorem 3.1 already implies thatone can obtain a solution of cost at most10.78ργ

∑

j,t txj,t, because settingze,t =∑

Q∈Tγt:e∈QzQ,t yields

a solution(x, z) that satisfies∑

e deze,t ≤ ργt and all other constraints of (LP1).) We sketch a (randomized)rounding procedure that also works forMGL and yields improved guarantees. We may assume that eachzQ,t ∈ [0, 1]. At each timetℓ = 2ℓ, ℓ = ⌈log2 T+ 4 log2 m⌉, we select at most⌈wtℓ⌉ trees fromTbtℓ ,picking eachQ ∈ Tγtℓ with probability zQ,tℓ . (We can always do this (efficiently) since for every timet,

the polytopez ∈ [0, 1]Tbt :∑

Q zQ,t ≤⌈

∑

Q zQ,t

⌉

is integral.) We take the union of all these trees.

Note that expected cost of the resulting subgraph is at mostwtγtℓ ≤ ργtℓ. We “Eulerify” the resultingsubgraph to obtain a tour fortℓ of cost at most2ργtℓ, and concatenate these tours. The probability thatj is not visited (or covered) by the tour fortℓ is at most1 −

∑

t≤tℓxj,t, which implies that with high

17

probability, we obtain a tour spanning all nodes. Lettingτj = tj(

23

)

, the expected latency-cost ofj is atmost2ργ

(

2τj + 2τj∑

k≥0(23 )

k)

≤ 16ργτj .

To prove part (ii), we adopt a rounding procedure that again utilizes Lemma 3.2, and yields the statedbound (deterministically). Recall thatτj(α) denotes theα-point of j. Let Dt(α) = j : τj(α) ≤ t. Foranyα ∈ (0, 1), and timet, we now show to obtain a tour spanningDt(α) ∪ r of cost at most2ργtα . Wecan then proceed as in the rounding procedure for Theorem 3.1to argue that, for the tours we obtain, thequantity

∑

iCi(Ni − Ni−1) appearing in Lemma 3.2 is bounded by2ργ∑

jtj(α)α . Hence, choosingα as

before according to the distributionq(x) = 2x and taking expectations, we obtain a solution with the statedcost.

LetK be such thatKα, KzP,tP∈Pγt are integers. For eachP with zP,t > 0, we createKzP,t copiesof each edge onP , and direct the edges away fromr. Let AP,t denote the resulting arc-set. Note thatin At :=

⊎

P :zP,t>0 AP,t, every nodej ∈ D has in-degree at least its out-degree, and there areKα arc-disjoint paths fromr to j for eachj ∈ Dt(α). So applying Theorem 2.6 in Bang-Jensen et al. [4], one canobtainKα arc-disjoint out-arborescences rooted atr, each containing all nodes ofDt(α). Thus, if we pickthe cheapest such arborescence and “Eulerify” it, we obtaina tour spanningDt(α) ∪ r of cost at most2 ·Kργt · 1

Kα = 2ργtα .

In Appendix A, we prove an analogue of Lemma 3.3 forMGL. Combined with part (i) of Theorem LP2P ,this shows that (even) a bicriteria approximation for “group orienteering” yields an approximation forMGLwhile losing a constant factor.

4 Extensions

We now consider various well-motivated extensions ofMLUFL, and show that our LP-based techniques andalgorithms are quite versatile and extend with minimal effort to yield approximation guarantees for thesemore generalMLUFL problems. Our goal here is to emphasize the flexibility afforded by our LP-basedtechniques, and we have not attempted to optimize the approximation factors.

Monotone latency-cost functions with bounded growth, and higher Lp norms. Consider the general-ization of MLUFL, where we have a non-decreasing functionλ(.) and the latency-cost of clientj is givenby λ(time taken to reach the facility servingj); the goal, as before, is to minimize the sum of the facility-opening, client-connection, and client-latency costs. Say thatλ has growth at mostp if λ(cx) ≤ cpλ(x)for all x ≥ 0, c ≥ 1. It is not hard to see that for concaveλ, we obtain the same performance guaranteesas those obtained in Section 2 (forλ(x) = x). So we focus on the case whenλ is convex, and obtain anO(

max(p log2 n)p, p log n logm)

-approximation algorithm for convex latency functions of growthp. Asa corollary, we obtain anO

(

p log nmaxlog n, logm)

-approximation forLp-MLUFL, where we seek tominimize the facility-opening cost + client-connection cost + theLp-norm of client-latencies.

Theorem 4.1 There is anO(

max(p log2 n)p, p log n logm)

-approximation algorithm forMLUFL withconvex monotonic latency functions of growth (at most)p.

Proof : We highlight the changes to the algorithm and analysis in Section 2.1. We again assume thatT =poly(m) for convenience. This assumption can be dropped by proceeding as in Theorem 2.8; we do afterproving the theorem. The objective of (P) now changes tomin

∑

i,t fiyi,t+∑

j,i,t

(

cij+λ(t))

xij,t. Theonly

change to Algorithm 1 is that we now definetℓ = min2ℓ/p,T and setN :=⌈

p log2(21/pτmax) + 4 log2 m

⌉

=O(p logm). DefineLcostj =

∑

i,t λ(t)xij,t. Note that we now haveλ(τj) ≤ 12Lcostj. Defineℓj to be thefirst phaseℓ such thattℓ ≥ τj. Let the random variablePj be as defined in Lemma 2.5. The failure probabil-ity of the algorithm is again at most1/ poly(m). The facility-cost incurred inO(p log n logm)

∑

i,t fiyi,t,

18

and the connection cost of clientj is at most4C∗j . We generalize Lemma 2.5 below to show thatE

[

Lj

]

=

O(

(p log2 n)p)

λ(tℓj ) ≤ O(

(2p log2 n)p)

· λ(τj), which yields the desired approximation. We have

Lj ≤ λ(

∑

ℓ≤Pj

d(Tourℓ))

≤ λ(

O(log2 n)∑

ℓ≤Pj

tℓ)

≤ O(log2p n)λ(

∑

ℓ≤Pj

tℓ)

,

soE[

Lj

]

≤ O(log2p n)[

λ(∑

ℓ≤ℓjtℓ)

+∑

k≥1 Pr[Pj ≥ ℓj + k]λ(∑

ℓ≤ℓj+k tℓ)

]

. Now,Pr[Pj ≥ ℓj + k] ≤(

49

)k,∑

ℓ≤ℓj+k tℓ ≤ tℓj ·2k/p

1−2−1/p , and(21/p − 1) ≥ ln 2p . Plugging these in gives,

E[

Lj

]

= O(log2p n) ·2

(21/p − 1)p· λ(tℓj ) ·

∑

k≥0

(4

9

)k2k = O

(

(p/ ln 2)p log2p n)

λ(tℓj).

Removing the assumption T = poly(m) in Theorem 4.1. As in the case of Theorem 2.8, to drop the as-sumption thatT = poly(m), we (a) solve the LP considering only times inTS = T0, . . . ,Tk (whereTr =⌈(1 + ǫ)r⌉); and (b) settℓ = TS(L·2ℓ/p) and the number of phases toN :=

⌈

p log2(21/pτmax/L) + 4 log2 m

⌉

,whereL = (

∑

j,i,t txij,t)/m. Note thatN = O(p logm), andλ(L) ≤ (∑

j,i,t λ(t)xij,t)/m = (∑

j Lcostj)/m,which shows that the expected latency cost incurred for clientsj with τj ≤ t0 is at most

∑

j Lcostj .

Corollary 4.2 One can obtain anO(

p log nmaxlog n, logm)

-approximation algorithm forLp-MLUFL.

Proof : As is standard, we enumerate all possible values of theLp-norm of the optimal client latencies inpowers of2, losing potentially another factor of2 in the approximation factor. To avoid getting into issuesabout estimating the solution-cost for a given guess (sinceour algorithms are randomized), we proceed asfollows. For a given guessLat, we solve (P) modifying the objective to bemin

∑

i,t fiyi,t +∑

j,i,t cijxij,t,and we adding the constraint

∑

j,i,t tpxij,t ≤ Latp. Among all such guesses and corresponding optimal

solutions, let(x, y, z) be the solution that minimizes∑

i,t fiyi,t +∑

j,i,t cijxij,t + Lat. Let OPT denotethis minimum value. Note thatOPT ≤ 2O∗, whereO∗ is the optimum value of theLp-MLUFL instance.We apply Theorem 4.1 (withλ(x) = xp) to round(x, y, z). LetF , C, andL =

∑

j Lj, denote respectivelythe (random) facility-opening, connection-, and latency-cost of the resulting solution. The bounds in Theo-

rem 4.1 imply thatE[

(∑

j Lj)1/p

]

≤(

E[∑

j Lj

])1/p≤ O(p log2 n)Lat, which combined with the bounds

onE[

F]

andE[

C]

, shows that the expected total cost isO(

p log nmaxlog n, logm)

OPT .

We obtain significantly improved guarantees for relatedMLUFL, metric uniformMLUFL, andML with(convex) latency functions of growthp. For relatedMLUFL and ML, the analyses in Sections 2.2 and 3directly yield anO

(

2O(p))

-approximation guarantee since for both problems, we can (deterministically)bound the delay of clientj by O

(

α-point of j)

(for suitableα). For metric uniformMLUFL, we obtainan O(1)-approximation bound as a consequence of Theorem 2.12: thisfollows because one can deviseanO(1)-approximation algorithm for the zero-facility-cost version of the problem by adapting the ideasused in [3]. Thus, we obtain anO(1)-approximation for theLp-versions of related-MLUFL, metric uniformMLUFL, andML.

k-route MLUFL with length bounds. All our algorithms easily generalize tok-route length-boundedMLUFL, where we are given a budgetB and we may use (at most)k paths starting atr of (d-) length atmostB to traverse the open facilities and activate them. This captures the scenario where one can usekvehicles in parallel, each with capacityB, starting at the root depot to activate the open facilities.Observethat withB =∞, we obtain a generalization of thek-traveling repairmenproblem considered in [14].

We modify (P) by settingT = B and setting the RHS of (3) tokt. In Algorithm 1, we now obtain atourTourℓ in phaseℓ of (expected length)O(log2 n)ktℓ. Each facilityi ∈ Tourℓ satisfies

∑

t≤tℓyi,t > 0,

19

so d(i, r) ≤ tℓ, and we may therefore divideTourℓ into k tours of length at mostO(log2 n)tℓ. Thus, weobtain the same guarantee on the expected cost incurred, andwe violate the budget by anO(log2 n)-factor,that is, we get a bicriteria

(

polylog, O(log2 n))

-approximation. Similarly, for relatedMLUFL andML, weobtain an

(

O(1), O(1))

-approximation. ForML, we may again use either (LP1) or (LP2P ): in both LPswe setT = B; in (LP1), we now have the constraint

∑

e deze,t ≤ kt, and in (LP2P ), the RHS of (5) isnow k. For metric uniformMLUFL, we modify (Unif-P) in the obvious way: we now have

∑

t yi,t ≤ k foreach timet, andt now ranges from 1 toB. We can again apply Theorem 2.12 here to obtain a (unicriteria)O(1)-approximation algorithm: for the zero-facility-location problem (where we may now “open” at mostkB facilities), we can adapt the ideas in [3] to devise anO(1)-approximation algorithm.

Finally, these guarantees extend to latency functions of bounded growth (in the same way that guar-antees forMLUFL extend to the setting with latency functions). Thus, in particular, we obtain anO(1)-approximation algorithmfor theLp-norm k-traveling repairmen problem; this is thefirst approximationguarantee for this problem.

Non-uniform latency costs. We consider here the setting where each clientj has a (possibly different)time-to-cost conversion factorλj, which measuresj’s sensitivity to time delay (vs. connection cost); so thelatency cost of a clientj is now given byλjtj , wheretj is the delay faced by the facility servingj.

All our guarantees in Sections 2.2, 2.3, and 3 continue to hold in this non-uniform latency setting.In particular, we obtain a constant approximation guarantee for related metricMLUFL, metric uniformMLUFL, andML. Notice that the metric uniformMLUFL problem cannot now be solved via a reductionto the metric-UFL variant discussed in Section 2.3; however we can still use Corollary 2.13 to obtain a10.773-approximation. For generalMLUFL, it is not hard to see that our analysis goes through under thethe assumptionT = poly(m). However, the scaling trick used to bypass this assumption leads to an extraO(

log(λmaxλmin

))

factor in the approximation.

Recall that the scaling factorL (in the definition oftℓ) in Section 2.1 was defined as∑

j L∗j/m, where

L∗j =

∑

i,t txij,t. Now,L∗j =

∑

i,t λjtxij,t, and we setL =∑

j L∗j/

∑

j λj. One can again argue that the

expected latency-cost of each clientj is at mostλj ·O(log2 n) ·max

L, τj, so we incur anO(log2 n)-factorin the latency-cost. The number of phases, however, isN :=

⌈

log2(2τmax/L) + 4 log2 m⌉

, and2τmax/L =

O(τmax

∑j λj∑

j L∗j

) ≤ O(

mλmaxλmin

), which gives an extralog2(λmax/λmin) factor in the facility-opening cost.

Finally, as before, these guarantees also translate to thek-route length-bounded versions of our problems.

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A Proofs omitted from the main body

Proof of Theorem 2.9 : Consider aGST instance(

H = (V,E), r, dee∈E , Gj ⊆ V mj=1). Letn = |V |.We may assume thatH is the complete graph,d is a metric, and the groups are disjoint. We abbreivateρn,mto ρ below. LetQ denote the collection of all solutions to the path-variant of GST; that is,Q consists ofall paths starting atr that visit at least one node of each group. For a pathQ ∈ Q and a groupGj , defineQj to be the portion ofQ from r to the first node ofGj lying onQ. Let dj(Q) =

∑

e∈Qjde be the length

of Qj ; that is,dj(Q) is the latency of groupj along pathQ. Consider the following LP-relaxation for thepath-variant ofGST, and its dual. We have a variablexQ for everyQ ∈ Q indicating if pathQ is chosen.We useQ below to index the paths inQ.

min M (P’)

s.t.∑

Q

dj(Q)xQ ≤M ∀j

∑

Q

xQ ≥ 1

x ≥ 0.

max α (D’)

s.t.∑

j

λjdj(Q) ≥ α ∀Q (10)

∑

j

λj ≤ 1

α, λ ≥ 0.

22

(P’) has an exponential number of variables. But observe that separating over the constraints (10) in thedual involves solving anMGL problem. The minimum value (over allQ ∈ Q) of the LHS of (10) is theoptimal value of theMGL problem defined by(Gj , λj), where we seek to minimize the weighted sum ofclient latency costs. (This weighted group latency problemcan be reduced to the unweighted problem by“creating” λj copies of each groupGj (we can scale theλjs so that they are integral); equivalently (insteadof explicitly creating copies), one can simulate this copying-process in whatever algorithm one uses for(unweighted)MGL.) Thus, aρ-approximation algorithm forMGL yields aρ-approximate separation oraclefor (D’). Now, applying an argument similar to the one used byJain et al. [24] shows that one can use thisto (also) obtain aρ-approximate solution(x,M) to (P’).

We now use randomized rounding to round(x,M) and obtain a group Steiner tree of cost at mostO(logm)M . We pick pathQ independently with probabilitymin4 logm · xQ, 1. LetQ′ ⊆ Q denotethe collection of paths picked. Note that for every groupGj , we have

∑

Q∈Q:dj(Q)≤2M xQ ≥12 . So a

standard set-cover argument shows that with probability atleast1 − 1/m, for everyj, there is some pathQ(j) ∈ Q′ such thatdj(Q(j)) ≤ 2M . We may assume that

∑

Q xQ = 1, so Chernoff bounds showthat |Q′| = O(logm) with overwhelming probability. The group Steiner treeT consists of the union of

all theQ(j)j (sub)paths (deleting edges to remove cycles as necessary).Clearly, the cost ofT is at most

|Q′| · 2M = O(logm)M . Note thatT also yields a path of lengthO(logm)M starting atr and visiting allgroups, so the integrality gap of (P’) isO(logm).

Extension of Lemma 3.3 to MGL. Notice that we did not use anything specific to the minimum-latencyproblem in the proof, and so essentially the same proof also applies toMGL. Recall that inMGL, we havea setF of facilities and ad-metric onF ∪ r, and a collection ofm groupsGj ⊆ F. Analogous to(

LP2(a,b)P

)

, the LP-relaxation with path variables forMGL and its dual are as follows.

min∑

j,t

txj,t (LP’(a,b)P )

s.t.∑

t

xj,t ≥ 1 ∀j

∑

P∈Pbt

zP,t ≤ a ∀t

∑

P∈Pbt:Gj∩P 6=∅

zP,t ≥∑

t′≤t

xj,t′ ∀j, t

x, z ≥ 0.

max∑

j

αj − a∑

t

βt (LD’ P(a, b))

s.t. αj ≤ t+∑

t′≥t

θj,t′ ∀j, t

∑

j:Gj∩P 6=∅

θj,t ≤ βt ∀t, P ∈ Pbt (11)

α, β, θ ≥ 0.

The LP-relaxation(

LP’(a,b)T

)

with tree variables is obtained by replacingPbt with Tbt in (LP’(a,b)P ). Theorienteering problem that we need to solve now to separate over the constraints (11) isgroup orienteering:given a rewardθj,t for each groupGj , we want to determine if there is a path (or tree) rooted atr of length atmostbt such that the total reward of the groups covered by it is more thanβt. Given these changes, the proofthat one can obtain a feasible solution(x, y) to

(

LP’(a,b)T

)

of cost at most the optimal value of(

LP’(1,1)P

)

isas in the proof of Lemma 3.3.

23