1 / 60

New Algorithms for Disjoint Paths Problems

New Algorithms for Disjoint Paths Problems. Sanjeev Khanna University of Pennsylvania. Joint work with Chandra Chekuri Bruce Shepherd. Edge Disjoint Path Problem (EDP). Input: Graph G(V,E) , node pairs s 1 t 1 , s 2 t 2 , ..., s k t k Goal: Route a maximum # of s i -t i pairs using

jola
Download Presentation

New Algorithms for Disjoint Paths Problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related