Disjoint Paths and Related Routing Problems

1. Disjoint Paths and Related Routing Problems Sanjeev Khanna University of Pennsylvania

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. si ti EDP on Line Networks Independent set in interval graphs.

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

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 • Maximize number of pairs connected • Allow small capacity violation Approximation algorithm A • 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 Heuristic with a provably good performance

9. A Greedy Algorithm • Among the unrouted pairs, pick the pair that has the shortest path in the 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. [Chuzhoy, K ’05] For undirected graphs, O( log1/2-e n)-approximation is hard. The O(m1/2)-approximation is the best known in general.

13. 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

14. Recent Progress [Chekuri, K, Shepherd: STOC 04, FOCS 04, STOC 05] • O(log2 n) approximation for ANF in general undirected graphs. • 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.

15. Significance of Results • Qualitative and quantitative improvements: polynomial to poly-logarithmic ratios. • Problem structure much better understood. • New concepts and connections.

16. Rest of the Talk EDP in planar graphs • A fractional relaxation • High level algorithm • Well-linked sets and crossbars

17. A general paradigm for approx Formulation Integer Program Problem Relaxation Analysis, approx ratio Linear Program Solution Rounding

18. Multicommodity Flow Formulation (IP) P(i) : set of paths between si and ti P =P(1)[ P(2) ... [P(k) f(p) : 1 if flow on path p 2P, 0 otherwise xi: 1 if siti is routed, 0 otherwise maxåi xi s.t xi = åp 2P(i) f(p) 1 · i · k åp: e 2 p f(p)·1 e 2 E xi, f(p) 2{0,1}

19. Multicommodity Flow Formulation (LP) P(i) : set of paths between si and ti P =P(1)[ P(2) ... [P(k) f(p) : 1 if flow on path p 2P, 0 otherwise xi: 1 if siti is routed, 0 otherwise maxåi xi s.t xi = åp 2P(i) f(p) 1 · i · k åp: e 2 p f(p)·1 e 2 E xi, f(p) 2[0,1]

20. Randomized Rounding [Raghavan,Thompson ’87] Pick each si - ti pair with probability equal to the si-ti flow routed in the LP solution. If a pair is chosen for routing, pick one of its flow paths for routing, again with probability proportional to the flow on it. Can route an optimal number of pairs provided log n /log log n paths are allowed to share an edge!

21. Randomized Rounding A powerful algorithm design technique with many applications. But does not exploit the structure of the underlying graph while converting the fractional solution to an integral one. Any good rounding scheme based on the LP must allow at least two paths to share an edge [Garg, Vazirani, Yannakakis ’93].

22. How Good is this LP? tk tk-1 ti t3 t2 t1 sk si s3 s2 s1 sk-1 Gap holds for planar graphs

23. High Level Algorithm • Solve the LP relaxation. • Use LP solution to decompose input instance into a collection of well-linked instances. • Well-linked instances have nice properties – exploit them!

24. S V - S Well-linked Set Subset X iswell-linkedin 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|

25. Input instance: G, X, M G : underlying graph. X : {s1, t1, s2, t2, ..., sk, tk} is the terminal set M : a pairing 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

26. s1 s1 t1 t1 s2 s2 t2 t2 s3 s3 t3 t3 s4 s4 t4 t4 Examples Nota well-linked instance A well-linked instance

27. s1 t1 s2 t2 s3 t3 s4 t4 Advantage of Well-linkedness LP value does not depend on input pairing M. Claim:If X is well-linked, then for any pairing on X, LP value is W(|X|/log |X|). We have symmetry w.r.t. to every pairing.

28. 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

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

30. v Grids in Planar Graphs Theorem [Robertson, Seymour, Thomas’94]: If G is a planar graph with a well-linked set of size h, then G has a grid minorH of size W(h) as a subgraph. Gv Gv Grid minor is crossbar with congestion 2

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

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

33. 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 large crossbar structure. Use the crossbar to route a large fraction of the pairs.

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

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

36. Decomposition • Can achieve b = O(log2 k) in general graphs. • Improves to O(log k) for planar graphs. Decomposition based on LP solution. Recursive algorithm using separator algorithms. Need to work with approximate and weighted notions of well-linked sets.

37. Decomposition Algorithm Weighted version of well-linkedness • each v 2 X has a weight • weight determined by LP solution • weight of si and ti equal to xi the flow in LP soln • X is well-linkedimplies no sparse cut. • If sparse cut exists, break the graph into two pieces. • Recurse on each piece. • Final pieces determine the decomposition.

38. Summary • New algorithmic framework for routing problems. • New connections: • notion of well-linkedness • well-linkedness and grid minors • decomposition into well-linked instances • General graphs? • Need to show existence of large crossbars.

39. Thank You!

40. si ti Some special cases di, pi Knapsack problem!Single Edge ue = B Independent set in interval graphs ! Line network

41. Related Routing Problems • Node capacitated problems • node disjoint paths • all-or-nothing flow • Demand problems: • each pair siti has a demand di • edges/nodes have capacities

42. Approximation Algorithms Overcome intractability of various kinds • NP-hard problems • polynomial time solvable but need fast(er) solutions • small space • can see data only once or a few times (streaming) Estimates and approximations often suffice Explain performance of current heuristics, bounds, and relaxations: obtain new & improved heuristics

43. Approximability of Routing Problems Approximation bounds: - O(min(n2/3, m1/2)) in general graphs [K,S,KS,CK02] - O(1) in trees [GVY93], grids, grid like graphs [KT95a,b] - O(log n) in constant degree expanders [KS02] Hardness of approximation: - EDP, AN-MCF APX-hard [GVY94] - Recent result W((log n)1/2-e)-hard [CK05]

44. Analysis of the Greedy Algorithm X : Set of pairs routed by the greedy. O: Set of pairs routed by an optimal solution. Partition O n X into two sets: • O_1: pairs whose paths intersect with a path chosen by greedy. • O_2: all remaining pairs in O. Then |O| = |X| + |O_1| + |O_2|.

45. Analysis of the Greedy Algorithm Suppose we discard from greedy solution any routed pairs whose path length is more than B. Then |O1| · B |X| and |O2| · m/B. Combining, |O| ·(|X| + B|X| + m/B). Choose B = m1/2 to get |O| · 3 m1/2 X. Thus the Greedy gives an O(m1/2)-approximation.

46. Flows, Cuts, and Integer Flows maxflow-mincut theorem one pair s,t :single commodity max integer flow = max frac flow = min cut

47. Flow-cut gap thms[LR88 ...] ?? graph theory Flows, Cuts, and Integer Flows Multicommodity: several pairs NP-hard Solvable NP-hard max integer flow · max frac flow · min multicut

48. Back to Well-linked sets Claim: X is well-linked implies treewidth = W(|X|) Xwell-linked)G has grid minor H of size W(|X|) Q: how do we route M = (s1t1, ..., sktk) using H ?