Universal Approximations in Network Design

Rajmohan RajaramanNortheastern University

Based on joint work with Chinmoy Dutta, Lujun Jia, Guolong Lin,

Jaikumar Radhakrishnan, Ravi Sundaram, Emanuele Viola

A day in the life of a courier

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56

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1 2 3 4 5 6 7 8

1 20 15 25 40 35 10 102 40 10 35 30 20 153 30 20 15 10 204 10 20 25 205 10 35 356 25 357 108

1,2,3,6?2,3,4,5?

1,2,3,6: 20+15+15+30=90

2,3,4,5: 10+10+20+40=80

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1,2,3,6?

2,3,4,5?

1,2,3,6: 20+15+15+30=9020+15+15+30=90 20+40+15+20=952,3,4,5: 10+10+20+40=80 40+30+10+35=115

1 2 3 4 5 6 7 8

1 20 15 25 40 35 10 102 40 10 35 30 20 153 30 20 15 10 204 10 20 25 205 10 35 356 25 357 108

Stretch ≥ 115/80

Consider tour 1,2,3,4,5,6,7,8

A day in…a lazy courier

Is there a smart lazy courier?

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• Is there a single tour that is universally good?

• Given any subset of the cities, the restricted subtour (preserving the order) is a good approximation to the best tour for that subset

Universal TSP

• Given a metric space (V,d), design a tour T over V that minimizes

– Ts is the sub-tour of T induced by S– Cost(X) is length of tour X– OPT(S) is cost of an optimal tour for S€

maxS⊆V

Cost(TS )OPT(S)

Universal Steiner tree (UST)Candidate

spanning tree TSet S

Cost of inducedSubtree TS = 16

OPT(S) = 6

Stretch ≥ 16/6

Universal Steiner tree (UST)• Given: A weighted graph G over set V of vertices,

and a root vertex• Goal: Determine a spanning tree T of G that

minimizes

– Ts is the subtree of T induced by S and root– OPT(S) is cost of optimal tree connecting S to root– Cost(X) is the sum of weights of edges in X

• Graphical UST: T only has edges from G• Metric UST: Work with metric completion of G• Graphical UST at least as hard as metric UST

€

maxS⊆V

Cost(TS )OPT (S)

Stretch of T

Universal approximations framework

• Universal version of optimization problem has two additional notions

• Sub-instance relation ≤• Restriction function R: Takes a solution S for

instance I, a sub-instance I’ of I, and returns a solution R(S,I,I’) for I’

• Goal: For given instance I, determine a solution T for I that minimizes

€

maxI '≤I

Cost(R(S,I,I'))OPT(I')

Universal set cover

• Given set V of elements, collection C of subsets of V, determine f: V C such that – For all x in V, f(x) contains x, and – The following stretch is minimized

€

maxS⊆V

Cost( f (S))OPT(S)

Motivation• Optimization under uncertainty

– Universal solutions are robust against adversarial inputs

• Aggregation tree in a sensor network– Data is being generated at several sensors, and

aggregation queries arrive at a sink – Setting up aggregation trees dynamically as

queries and data change may be expensive– Universal Steiner trees provide good

approximations for all query and update patterns • Universal solutions are differentially private

[Bhalgat-Chakrabarty-Khanna 11]

Universal approximation resultsUpper bound Lower bound

UTSPO(log4n/loglogn) [JLNRS 05]O(log2n) [GHR 06]O(log n) for doubling metrics

Ω(log n) [GKSS 10, BCK 11]

Ω(log1/6n) Euclidean

Metric USTO(log4n/loglogn) [JLNRS 05]O(log2n) [GHR 06]O(log n) for doubling metricsO(log n) for planar [KHL 06]

Ω(log n) [JLNRS 05]

Ω(logn/loglog n) Eucl.

Graph UST [DRRSV 11]

for O(1) doubling dimSame as for metric UST

USCWeighted: O(√nlogn) [JLNRS 05]

Unweighted: O(√n)Ω(√n) [JLNRS 05] €

2 ˜ O (log3/4 n )

€

2 ˜ O ( log n )

The roadmap• Landscape around universal approximations• Universal Steiner trees

– Bounded locally consistent partitions– Metric UST– Graphical UST– Lower bound

• Concluding remarks and open problems

The “universal” landscape• O(log n)-stretch universal TSP for the Euclidean plane

[Platzman-Bartholdi 89]• Simultaneous approximations for single-sink buy-at-

bulk– Given a graph, demands to be routed to a sink, cost for each

edge, route demands to minimize total cost– A single tree is a simultaneous O(1)-approximation for all

concave cost functions [Goel-Estrin 03,…,Goel-Post 10]• Tree decompositions

– [Fakcharoenphol-Rao-Talwar 03] yields metric tree whose expected stretch for each set is O(log n)

– O(log n loglog n) using [Elkin-Emek-Spielman-Teng 06, Abraham-Bartal-Neiman 09] distribution over spanning trees

– The cut-based decompositions of [Räcke 02,08] also aim for a distribution over trees or tree with prob. embedding

The “universal” landscape• Oblivious routing and network design

– Given graph, source-sink pairs, and per-edge routing cost, determine routes that are oblivious to demand pairs and cost function

– O(log2n)-approximation for sub-additive cost functions – [Räcke 02, Harrelson-Hildrum-Rao 03, Gupta-Hajiaghayi-

Räcke 06] • A priori approximations [Schalekamp-Shmoys 08]

– For TSP, set of vertices visited drawn from a probability distribution

• Set covering with eyes closed– Determine a single mapping of elements to sets to minimize

expected cost of covering random element subset– [Grandoni-Gupta-Leonardi-Miettinen-Sankowski-Singh 08]

Universal Steiner tree (UST)• Given: A weighted graph G over set V of

vertices, and a root vertex • Goal: Determine a spanning tree T of G that

minimizes

– Ts is the subtree of T induced by S and root– OPT(S) is cost of an optimal tree connecting S to

root– Cost(X) is the sum of weights of edges in X

€

maxS⊆V

Cost(TS )OPT (S)

Stretch of T

What does a good UST look like?Candidate

spanning tree T

What does a good UST look like?• At each distance level, T

provides a clustering of G• Given tree T, adversary

identifies set S such that– S is “well-separated” in T– S is “close” in G

• To avoid this, UST should cluster nodes so that – Each node’s neighborhood

does not intersect too many clusters

– Otherwise, adversary will select several nodes from this neighborhood lying in different clusters

Bounded locally consistent partitions• A partition of the metric

space with the following properties:– Diameter of every cluster in

partition is at most αR– Every R-ball intersects β

clusters• Every metric space has an

(O(log n),O(log n),R)-partition for every R– Sparse partitions [Awerbuch-

Peleg 90], [Peleg 00]β = 4

Hierarchical partitions• A collection of partitions Pi

with the following properties:– Partition: Pi is an (α,β,Ri)-

partition– Hierarchy: Pi is a refinement of

Pi+1

– Root padding: Cluster in Pi containing root contains ball of radius Ri around root

• Every metric space has a hierarchical (O(logn), O(logn),O(logn))- partition

A metric UST algorithm [JLNRS 05]• Compute a hierarchical (O(log n),O(log

n),O(log n))-partition• For each level i, from lowest to highest:

– For each level i cluster:• Select leader from leaders of its constituent level i-1

clusters• connect level i leader to level i-1 leaders

• Root is always leader of its clusters

Proof sketch for stretch• To prove: For every set S,

Cost(TS) is at most polylog(n) times OPT(S)

• For a level j, cost in UST is O(njlogj+1n):– nj is the number of level-j

ancestors of nodes in S• Main Lemma:

– If Pj is a maximal set of nodes in S pair-wise separated by logj-1n

– Then nj = O(|Pj| log n) • Cost(OPT(S)) is Ω(|Pj|logj-1n)• Cost at level j in UST is thus

O(log3n)Cost(OPT(S))

Bounding the cost at a level

Proof sketch of Main Lemma:• Any node’s ancestor at level j

is within O(logjn) cost of node• Therefore, O(logjn)-ball

around the ancestors of Pj at level j covers all nj ancestors of S at level j

• By partitioning scheme, it follows that nj is O(|Pj|log n)

Pj is maximal set of nodes in S pair-wise separated by logj-1nnj is the number of nodes at level j of induced tree We have nj = O(|Pj| log n)

Improved bounds for special metrics• For doubling metrics, the UST algorithm

achieves a stretch of O(log n)– Hierarchical (O(1),O(1),O(1))-partition

• Doubling metrics include Euclidean metrics as well as growth-restricted metrics

An O(log2n)-stretch metric UST• Gupta-Hajiaghayi-Räcke 06• α-padding: A node v is α-padded in a

hierarchical decomposition if– At level i, the ball of radius α2i around v is fully

contained within its cluster at level i• Theorem: For any v, in any tree drawn from

the [FRT 03] distribution, probability that v is Ω(1/logn)-padded is at least 3/4

An O(log2n)-stretch metric UST• Simple metric UST construction:

– Sample O(log n) trees from the FRT distribution– For each vertex v select a tree where v is

Ω(1/logn)-padded– In each tree, build the sub-tree induced by the

root and vertices that selected the tree (using metric completion)

– Return the union of the O(log n) sub-trees computed above

• O(log2n) stretch

Challenges for Graphical UST• Bounded locally consistent partition:

– Partition G into clusters of strong diameter at most αR

– Each R-ball intersects at most β clusters– How small can α and β be? – Open: Is (polylog(n), polylog(n),1)-partitioning

achievable?• Lemma (Necessity): If σ-stretch achievable

for graphical UST, then (σ,σ2,R)-partition exists for all R.

Challenges for Graphical UST• Hierarchical partition:

– Unlike in the metric case, cannot simply elect leaders and connect directly

– Connecting lower level partitions arbitrarily may introduce huge blowup in costs

• In the [GHR 06] approach:– Can replace the O(log n) FRT trees by the spanning

trees drawn from [EEST 05] distribution– Not clear how to combine paths drawn from these

trees into a single spanning tree

Graphical UST construction• Construct (2Õ(√logn), 2Õ(√logn),R)-partitions

– (O(1),O(1),R) for doubling graphs• Convert to a hierarchical partitioning:

– (2Õ(√logn),2Õ(√logn),2Õ(√logn)) for general graphs– (O(1), O(1), O(log2n)) for doubling graphs

• Build UST from hierarchical partition:– Connect lower-level trees using shortest paths– Invoke properties of partitioning to bound stretch

• for general graphs and 2Õ(√logn) for doubling graphs

• [Dutta-Radhakrishnan-R-Sundaram-Viola 11]€

2 ˜ O (log3/4 n )

Lower bound for UST• Every algorithm for on-line Steiner trees

over n nodes has a competitive ratio of (log n) for metrics [Imase-Waxman 91] (log n/loglog n) for Euclidean metrics [Alon-

Azar 92] • Any UST for an n-node metric space with

stretch s(n) can be transformed into an on-line algorithm with competitive ratio of s(n)– Consequence: Every UST has a stretch of (log

n) for n-node metrics, (log n/loglog n) for Euclidean metrics

Complexity of universal problems

• For a given terminal set S:– Finding OPT(S) is NP-hard– Poly-time O(1)-approximations known (Minimum spanning

tree,…,[])• For a candidate UST, finding the worst-case set is

NP-hard• Finding whether there exists a UST with stretch at

most σ is coNP-hard• Universal problems are “”-optimization problems

– The -quantification is over an exponential-sized domain– Lies in ∑2

– Open: is it ∑2-hard?

€

maxS⊆V

Cost(TS )OPT (S)

Open problems• Close the gaps for UTSP and metric UST

– Euclidean UTSP: Ω(log1/6n) vs O(log n)– UTSP: Ω(log n) vs O(log2 n)– Metric UST: Ω(log n) vs O(log2n)

• Is there a polylog(n)-stretch graphical UST?• Strong diameter partitions:

– Can we partition any graph into components of strong diameter polylog(n) such that each vertex has neighbors in polylog(n) components?

– [Peleg 00]• Universal approximations for other

optimization problems