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All-or-Nothing Multicommodity Flow

This research explores the All-or-Nothing Multicommodity Flow (AN-MCF) problem in optical networks, where the goal is to maximize the number of paired connections routed over a network with specific capacities. We delve into the complexity of unsplittable flow paths, approximation algorithms for edge-disjoint paths, and present clustering strategies to improve routing efficiency. By leveraging hierarchical graph decomposition techniques, we investigate the feasibility of routing multiple pairs while maintaining minimal congestion and optimizing flow distribution across the 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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