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All-or-Nothing Multicommodity Flow Chandra Chekuri Sanjeev Khanna Bruce Shepherd Bell Labs U. Penn Bell Labs

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Routing connections in networks 25 NY SE DE NY – SF 10 Gb/sec NY – SF 20 SE – DE 5 SF – DE 6 Core Optical Network

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Multicommodity Routing Problem Network – graph with edge capacities Requests: k pairs, (s i, t i ) with demand d i Objective: find a feasible routing for all pairs Optimization: maximize number of pairs routed

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All-or-Nothing Flow Problems Pair is routed only if all of d i satisfied Single path for routing: unsplittable flow (connection oriented networks) Fractional flow paths: all-or-nothing flow (packet routing networks) Integer flow paths: all-or-nothing integer flow (wavelength paths)

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Complexity of AN-Flow d i = 1 for all i Single path: edge disjoint paths problem (EDP) classical problem, NP-hard only polynomial approx ratios AN-MCF: APX-hard on trees approximation ?

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Approximating EDP/AN-MCF O(min(n 2/3,m 1/2 )) approx in dir/undir graphs (EDP/UFP) [Kleinberg 95, Srinivasan 97, Kolliopoulos-Stein 98, C-Khanna 03, Varadarajan-Venkataraman 04] EDP is (n 1/2 - )-hard to approx in directed graphs [Guruswami-Khanna-Rajaraman-Shepherd-Yannakakis 99] LP integrality gap for EDP is (n 1/2 ) [GVY 93] AN-MCF: APX-hard on trees [Garg-Vazirani-Yannakakis 93]

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Results In undirected graphs AN-MCF has an O(log 3 n log log n) approximation Polynomial factor to poly-logarithmic factor Approx via LP, integrality gap not large For planar graphs O(log 2 n log log n) approx Same ratios for arbitrary demands: d max · u min Online algorithm with same ratio

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LP Relaxation x i : amount of flow routed for pair (s i, t i ) max i x i s.t x i flow is routed for (s i,t i )1 · i · k 0 · x i · 11 · i · k

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A Simple Fact Given AN-MCF instance: all d i = 1 Can find (OPT) pairs such that each pair routes 1/log n flow each How? rand rounding of LP and scaling down Problem: we need pairs that send 1 unit each

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Nice Flow Paths Suppose all flow paths use a single vertex v v s1s1 s2s2 s3s3 s4s4 t1t1 t2t2 t3t3 t4t4

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Routing via Clustering v cluster has log n terminals cluster induces a connected component clusters are edge disjoint

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Clustering Finding connected edge-disjoint clusters? G is connected: use a spanning tree for a rough grouping of terminals New copy of G for clustering: congestion 2 1 for clustering, 1 for routing Congestion 1 using complicated clustering

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How to find nice flow paths? Algorithmic tool: Racke’s hierarchical graph decomposition for oblivious routing [Räcke02]

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Räcke’s Graph Decomposition Represent G as a capacitated tree T leaves of T are vertices of G internal node v: G(v) is induced graph on leaves of T(v) v

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Räcke’s Result T is a proxy for G For all D c*(D,G) · c(D,T) · (G) c*(D,G) Routing in T is unique (G) = O(log 3 n) [Räcke 02] (G) = O(log 2 n log log n) [Harrelson-Hildrum-Rao 03]

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Routing details With each v there is distribution v on G(v) s.t i 2 G(v) v (i) = 1 s distributes 1 unit of flow to G(v) according to v t distributes 1 unit of flow to G(v) according to v v s t

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Back to Nice Flow Paths v s1s1 s2s2 s3s3 s4s4 t1t1 t2t2 t3t3 t4t4 G(v), v s1s1 s2s2 s3s3 s4s4 t1t1 t2t2 t3t3 t4t4 X(v): pairs with v as their least common ancestor (lca) Routing in TRouting in G

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Algorithm Find set of pairs X that can be routed in T (use tree algorithm [GVY93,CMS03] ) Each pair (s i,t i ) in X has a level L(i) Choose level L* at which most pairs turn Route pairs independently in subgraphs at L* L* v

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Algorithm cont’d v at L*, X(v) pairs in X that turn at v Can route 1/ (G) flow for each pair in X(v) using nice flow paths Use clustering to route X(v)/ (G) pairs Approx ratio is (G) depth(T) = O(log 3 n log log n)

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Open Problems Improve approximation ratio What is integrality gap of LP ? No super-constant gap known Extend ideas to EDP Recent result: Poly-log approximation for EDP/UFP in planar graphs with congestion 3

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