Advances in Edge-Disjoint Path Algorithms: New Strategies and Approximation Techniques

1. New Algorithms for Disjoint Paths Problems Sanjeev Khanna University of Pennsylvania Joint work with Chandra ChekuriBruce Shepherd

2. Edge Disjoint Path Problem (EDP) Input: Graph G(V,E), node pairs s1t1, s2t2, ..., sktk Goal: Route a maximum # of si-ti pairs using edge-disjoint paths t3 s4 t4 s3 s2 t1 s1 t2

3. Edge Disjoint Path Problem (EDP) Input: Graph G(V,E), node pairs s1t1, s2t2, ..., sktk Goal: Route a maximum # of si-ti pairs using edge-disjoint paths t3 s4 t4 s3 s2 t1 s1 t2

4. u1 u4 1 1 1 1 u4 u1 u2 u3 u2 u3 Matching in G EDP in Star EDP on Stars And vice versa.

5. Two Pair Problem Input: Graph G(V,E) and two pairs s1t1, s2t2. Goal: Can we simultaneously route s1 to t1 and s2 to t2 in an edge-disjoint manner?

6. Two Pair Problem Input: Graph G(V,E) and two pairs s1t1, s2t2. Goal: Can we simultaneously route s1 to t1 and s2 to t2 in an edge-disjoint manner? • NP-hard if G is a directed graph [Fortune, Hopcroft, Wylie’80]. • Polynomial-time solvable for any constant number of pairs if G is undirected [Roberston, Seymour’88].

7. Routing Problems • Related problems: • node disjoint paths • each pair si-ti has a demand di and edges/nodes have capacities • Fundamental to combinatorial optimization • Applications to VLSI, network design and routing, resource allocation & related areas • Related to significant theoretical advances

8. Coping with Hardness Settle for sub-optimal solutions: • route only a fraction of the pairs that can be routed in an optimal solution • allow for small violations of edge capacities Approximation algorithmA • runs in polynomial time • approximation ratio: how good is A • approx ratiob)A(I) ¸ OPT(I)/b for all I • Would like b to be as small as possible

9. A Greedy Algorithm • Among the unrouted pairs, pick the pair that has the shortest path in the current graph. • Route this pair and remove all edges on the path from the graph. Clearly gives an edge-disjoint routing. How good is this algorithm?

10. Analysis of the Greedy Algorithm n: # of vertices m: # of edges Fix an optimal solution, say, OPT. If each greedy path is at most m1/2 edges long, it destroys at most m1/2 paths in OPT. Suppose at some point, a path chosen by greedy is longer than m1/2. Since there are only m edges, OPT can chose at most m1/2 paths from here on. So greedy gives an O(m1/2)-approximation.

11. A Bad Example Greedy chooses the red path and none of the blue pairs can be routed as a result.

12. Surely, we could do better ... Not if the graph is directed! [Guruswami, K, Rajaraman, Shepherd, Yannakakis ’99] It is NP-hard to get an O(m1/2 - e) approximation for directed graphs for any e > 0.

13. Surely, we could do better ... Not if the graph is directed! [Guruswami, K, Rajaraman, Shepherd, Yannakakis ’99] It is NP-hard to get an O(m1/2 - e) approximation for directed graphs for any e > 0. [Chuzhoy, K ’05] For undirected graphs, O( log1/2-e n)-approximation is hard. (Builds on [Andrews, Zhang ’05] .)

14. Surely, we could do better ... Not if the graph is directed! [Guruswami, K, Rajaraman, Shepherd, Yannakakis ’99] It is NP-hard to get an O(m1/2 - e) approximation for directed graphs for any e > 0. [Chuzhoy, K ’05] For undirected graphs, O( log1/2-e n)-approximation is hard. (Builds on [Andrews, Zhang ’05] .) The O(m1/2)-approximation is the best known in general (as a function of m).

15. All-or-Nothing Flow Prob (ANF) Input: Graph G(V,E), node pairs s1t1, s2t2, ..., sktk. Goal: Route a maximum # of si-ti pairs such that each routed pair has one unit of flow. s1 s1 s2 s2 1/2 1/2 1/2 t2 t1 t2 t1 1/2

16. Recent Progress [Chekuri, K, Shepherd: ’04 and ’05] • O(log2 n) approximation for ANF in undirected graphs. • [Chuzhoy, K ’05]O( log1/2-e n)-approximation is hard.

17. Recent Progress [Chekuri, K, Shepherd: ’04 and ’05] • O(log2 n) approximation for ANF in undirected graphs. • [Chuzhoy, K ’05]O( log1/2-e n)-approximation is hard. • O(logn) approximation for EDP in planar undirected graphs when up to two paths can share an edge.

18. Recent Progress [Chekuri, K, Shepherd: ’04 and ’05] • O(log2 n) approximation for ANF in undirected graphs. • [Chuzhoy, K ’05]O( log1/2-e n)-approximation is hard. • O(logn) approximation for EDP in planar undirected graphs when up to two paths can share an edge. • Similar results for the node-disjoint versions as well as versions with arbitrary demands and capacities.

19. Recent Progress [Chekuri, K, Shepherd: ’04 and ’05] • O(log2 n) approximation for ANF in undirected graphs. • [Chuzhoy, K ’05]O( log1/2-e n)-approximation is hard. • O(logn) approximation for EDP in planar undirected graphs when up to two paths can share an edge. • Similar results for the node-disjoint versions as well as versions with arbitrary demands and capacities. Previous algorithms had W(n1/2) approximation ratio.

20. Rest of the Talk • EDP in planar graphs • A fractional relaxation • A new framework for routing problems • Well-linked sets and crossbars • Routing using crossbar structures • EDP with congestion in general graphs

21. Multicommodity Flow Formulation (LP) • Routing is relaxed to be a flow from si to ti. • A pair can be routed for a fractional amount. xi: fraction of si-ti flow that is routed. Maxåi xi s.t. 8e total flow through e· 1. 0 · xi· 1.

22. Randomized Rounding [Raghavan and Thompson ’87] • For each pair (si,ti), decompose the flow xi2[0,1] into a collection of flow paths. • Decide to route pair (si,ti) with probability xi. • If yes, choose an si-ti flow path with probability proportional to the flow on it. O(n1/c)-approximation if we allow up to c paths to use an edge.

23. How Good is this LP? [GVY ’93] tk tk-1 ti t3 t2 t1 sk si s3 s2 s1 sk-1

24. How Good is this LP? [GVY ’93] tk tk-1 ti t3 (n1/2)Lower Bound t2 t1 sk si s3 s2 s1 sk-1 Gap holds for planar graphs

25. How Good is this LP? [GVY ’93] tk tk-1 ti t3 (n1/2)Lower Bound t2 O(n2/3) Upper Bound t1 sk si s3 s2 s1 sk-1 Gap holds for planar graphs

26. A New Framework • Solve the LP relaxation. • Use LP solution to decompose input instance into a collection of instances with special structure, called well-linked instances. • Well-linked instances have special properties; use them for routing!

27. High-level Algorithm for Planar Graphs • Solve the LP relaxation. • Use LP solution to decompose input instance into a collection of well-linked instances. • Well-linkedplanar instances have crossbars, use them for routing! Assume w.l.o.g. input graph to be bounded degree.

28. S V - S Well-linked Set Subset X iswell-linked in G if for every partition (S,V-S) , # of edges cut is at least # of X vertices in smaller side. For all S ½ V with |S Å X| · |X|/2, |d(S)| ¸ |S Å X|

29. Input instance: G, X, M G : underlying graph. X : {s1, t1, s2, t2, ..., sk, tk} is the terminal set M : a matching on X ,namely, (s1,t1), (s2,t2) ... (sk,tk) that needs to be routed in G. Instance of EDP

30. Input instance: G, X, M G : underlying graph. X : {s1, t1, s2, t2, ..., sk, tk} is the terminal set M : a matching on X ,namely, (s1,t1), (s2,t2) ... (sk,tk) that needs to be routed in G. X is well-linked in G. Well-linked Instance of EDP

31. s1 s1 t1 t1 s2 s2 t2 t2 s3 s3 t3 t3 s4 s4 t4 t4 Examples Not a well-linked instance A well-linked instance

32. Crossbars H=(V,E) is a cross-bar with respect to an interfaceI µ V if any matching on I can be routed using edge-disjoint paths. Ex: a complete graph is a cross-bar with I=V H

33. s1 s4 s5 t5 t1 t3 t4 s2 s3 t2 Grids as Crossbars First row is interface

34. v Grids in Planar Graphs Theorem[Robertson, Seymour, Thomas ’94]: If G is a planar graph with a well-linked set of size k, then G has a grid minorH of size W(k) as a subgraph. Gv Gv Grid minor is a crossbar with congestion 2 [Kleinberg ’96]: uses it for half disjoint paths.

35. Routing pairs in X using H H X

36. Routing pairs in X using H Sink H A Single-Source Single-Sink Flow Computation X Source

37. Routing pairs in X using H Sink H Route X to I Source X

38. Routing pairs in X using H H Route X to I X

39. Routing pairs in X using H H Route X to I and use H for pairing up X

40. Several Technical Issues • H is smaller than X, so can pairs reach H? • What if X cannot reach H? • Can X reach interface of H without using edges of H? • Can H be found in polynomial time?

41. Routing Pairs to I Claim: If somesubsetA of terminals can reach I, then any subsetA’ of terminals with |A’| · |A|/2 can reach I. Use the fact that the terminals are well-linked.

42. Routing Pairs to I V - S S I A’ p edges

43. Routing Pairs to I V - S S A I A A’ p edges If |S Å A| ¸ |A|/2, then p ¸ |A|/2 ¸ |A’| since A can reach B.

44. Routing Pairs to I V - S S I A’ A A p edges If |(V-S) Å A| ¸ |A|/2, then p ¸ |A|/2 since terminals are well-linked.

45. Routing Pairs to I V - S S I A’ p ¸ |A’| edges Thus A’ can be routed to I. We can choose any |A’|/2 pairs to be routed to I.

46. Summarizing ... • Solve the LP relaxation. • Use LP solution to decompose input instance into a collection of well-linked instances. • Well-linked instances on planar graphs have a grid crossbar. Use it to route many pairs.

47. Can Route the Entire Matching • For EDP, suffices to route simply a constant fraction of pairs in the EDP instance. • Actually, we can route the entire matching with O(1) congestion.

48. Decomposition into Well-linked Instances G G1 G2 Gr Xi is well-linked in Gi åi |Xi| ¸ OPT/b

49. s1 s1 t1 t1 Example s2 t2 s2 t2 s3 t3 s4 t4 s3 t3 s4 t4

50. Decomposition • b = O(log2 n) in general graphs. •  = O(log n) for planar graphs. Decomposition based on LP solution.