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

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

3. 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

4. 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)

5. 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 ?

6. 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]

7. 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

8. 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

9. 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

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

11. Routing via Clustering v 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 decomposition for oblivious routing [Räcke02]

14. 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) 10 3 2 4 7 4 v

15. 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]

16. 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

17. 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

18. 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

19. 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)

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