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All-or-Nothing Multicommodity Flow

All-or-Nothing Multicommodity Flow. Chandra Chekuri Sanjeev Khanna Bruce Shepherd Bell Labs U. Penn Bell Labs. 5. 10. 6. 20. Routing connections in networks. NY – SF 10 Gb/sec NY – SF 20 SE – DE 5 SF – DE 6. SE. DE. 25. NY. Core Optical Network.

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All-or-Nothing Multicommodity Flow

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

  2. 5 10 6 20 Routing connections in networks NY – SF 10 Gb/sec NY – SF 20 SE – DE 5 SF – DE 6 SE DE 25 NY Core Optical Network

  3. Multicommodity Routing Problem • Network – graph with edge capacities • Requests: k pairs, (si, ti)with demanddi Objective:find a feasible routing for all pairs Optimization: maximize number of pairs routed

  4. All-or-Nothing Flow Problems Pair is routed only if allof disatisfied 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)

  5. Complexity of AN-Flow di = 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 ?

  6. Approximating EDP/AN-MCF O(min(n2/3,m1/2)) approx in dir/undir graphs (EDP/UFP)[Kleinberg 95, Srinivasan 97, Kolliopoulos-Stein 98, C-Khanna 03, Varadarajan-Venkataraman 04] EDPisW(n1/2 - e)-hard to approx in directed graphs [Guruswami-Khanna-Rajaraman-Shepherd-Yannakakis 99] LP integrality gap for EDP isW(n1/2) [GVY 93] AN-MCF: APX-hard on trees [Garg-Vazirani-Yannakakis 93]

  7. Results In undirected graphs AN-MCF has an O(log3 n log log n) approximation Polynomial factor to poly-logarithmic factor Approx via LP, integrality gap not large For planar graphs O(log2 n log log n) approx Same ratios for arbitrary demands: dmax· umin Online algorithm with same ratio

  8. LP Relaxation xi: amount of flow routed for pair (si, ti) max åi xi s.t xi flow is routed for (si,ti)1 ·i ·k 0·xi·11·i ·k

  9. A Simple Fact Given AN-MCF instance: all di= 1 Can find W(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

  10. s1 v t1 s2 s3 t2 t3 s4 t4 Nice Flow Paths Suppose all flow paths use a single vertex v

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

  12. 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

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

  14. Räcke’s Graph Decomposition Represent G as a capacitated tree T 3 10 4 v 4 2 7 leaves of T are vertices ofG internal node v: G(v) is induced graph on leavesof T(v)

  15. Räcke’s Result T is a proxy for G For all D c*(D,G) · c(D,T)·a(G) c*(D,G) Routing in T is unique a(G) = O(log3 n) [Räcke 02] a(G) = O(log2 n log log n) [Harrelson-Hildrum-Rao 03]

  16. Routing details With eachvthere is distributionpvonG(v)s.t åi 2 G(v)pv(i)=1 s distributes 1 unit of flow to G(v) according to pv t distributes 1 unit of flow to G(v)according to pv v t s

  17. s1 v t1 s2 s3 t2 t3 s4 t4 Back to Nice Flow Paths X(v): pairs with v as their least common ancestor (lca) G(v),pv s1 t1 s2 s3 t2 t3 s4 t4 Routing in T Routing in G

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

  19. Algorithm cont’d • v at L* , X(v) pairs in X that turn at v • Can route 1/a(G) flow for each pair in X(v) using nice flow paths • Use clustering to route X(v)/a(G) pairs Approx ratio is a(G) depth(T) = O(log3 n log log n)

  20. 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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