Edge Disjoint Paths (EDP)

2. OPT = 1 Edge Disjoint Paths (EDP) • Input • Graph G (M edges, N nodes); • A set of demands, (si , ti); • Output • A subset of demands routed on edge-disjoint paths; • Maximize such a subset s t t s

3. OPT = 1 Edge Disjoint Paths (EDP) • Input • Graph G (M edges, N nodes); • A set of demands, (si , ti); • Output • A subset of demands routed on edge-disjoint paths; • Maximize such a subset s t t s • Hard problem • One of the first!! (Karp’s list)

4. OPT = 2 Congestion Minimization • Input • Graph G (M edges, N nodes); • A set of demands, (si , ti); • Output • Route all demands; • Minimize max number of demand routes per edge. s t t s

5. Edge Disjoint Paths with Congestion (EDPwC) • Input • Graph G (M edges, N nodes); • A set of demands, (si , ti); • Congestion parameter c; • Output • A subset of demands routed such that max congestion c • If ALG can route X/a demands with congestion c whenever OPT can route X demands with congestion 1 then • ALG is an a-approx with congestion c s t t s

6. Known Results • Undirected Graphs • EDP solvable in polytime if the number of demands is constant • Robertson-Seymour • Directed Graphs • NP-hard even for 2 demands • Fortune-Hopcroft-Wyllie

7. Known Results (Undirected Graphs) • Positive • N1/c - approx with congestion c • Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein • Negative • No log(1-)/(c+1) N- approx with congestion c • A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang • Open Question: • Is there a polylog(N)-approx with constant congestion?

8. Known Results (Undirected Graphs) • Positive • N1/c - approx with congestion c • Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein • Negative • No log(1-)/(c+1) N- approx with congestion c • A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang • Open Question: • Is there a polylog(N)-approx with constant congestion? Yes for planar graphs, “all-or-nothing” flow (Chekuri-Khanna-Shepherd)

9. Known Results (Directed Graphs) • Positive • N 1/c - approx with congestion c • Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein • Negative • No N Ω(1/c) - approx with congestion c • A-Zhang, Chuzhoy-Guruswami-Khanna-Talwar

10. Known Results (Undirected Graphs) • Positive • N1/c - approx with congestion c • Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein • Negative • No log(1-)/(c+1) N- approx with congestion c • A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang • Rao-Zhou conjecture: • There is a polylog(N)-approx with O(log log N) congestion

11. Results (Undirected Graphs) • Positive • N1/c - approx with congestion c • Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein • Negative • No log(1-)/(c+1) N- approx with congestion c • A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang • This talk: • There is a polylog(N)-approx with poly(log log N) congestion

12. Results (Undirected Graphs) • Positive • N1/c - approx with congestion c • Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein • Negative • No log(1-)/(c+1) N- approx with congestion c • A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang • This talk: • There is a polylog(N)-approx with poly(log log N) congestion

13. Results • Positive • N1/c - approx with congestion c • Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein • Negative • No log(1-)/(c+1) N- approx with congestion c • A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang • This talk: • There is a log61(N)-approx with (log log N)6 congestion

14. Talk Outline • Describe some previous work on EDP • Describe Rao-Zhou and Räcke • Rao-Zhou: EDP in graphs with large min-cut • Räcke: hierarchical graph decompositions • Merge previous analyses to get EDPwC result

15. EDPwC in Graphs with Short Paths • Fractional routing • If we allow fractional paths, problem is a linear program • Known result #1 • If fractional path length is polylog(N), answer to open question is “yes” • Why? • Use randomized rounding. Each edge is dependent on polylog(N) other edges • Apply Lovasz Local Lemma 1/2 s t 1/2 1/2 t s 1/2 • Open Question: • Is there a polylog(N)-approx with constant congestion?

16. Graph Expansion • Out-degree • Let out(S) = set of edges with one endpoint in S • Abuse: out(S) = | out(S) | • Expander • Graph G is an expander if • out(S) / |S| • is large whenever S is small

17. EDPwC in Expanders • Known result #2 • Answer to open question in expanders is “yes” • e.g. Broder-Frieze-Upfal, Kolman-Scheideler • Even better, can connect polylog(N) fraction of ANY set of terminals using constant congestion • Why? • Using random walks, can connect any pair of terminals with paths of polylog(N) length • Use known result #1

18. Rao-Zhou • Known result #3 • (Rao-Zhou) Answer to open question is “yes” if min-cut in graph has size polylog(k) ---- ( k = # source-demand pairs) • Open Question: • Is there a polylog(N)-approx with constant congestion?

19. Rao-Zhou Analysis • Why? • (Khandekar-Rao-Vazirani) Can build expander on | T | terminals using polylog( | T | ) bipartite matchings

20. Rao-Zhou Analysis • Application to EDPwC • Use max-flows to find matchings between terminals • How to bound congestion from max-flows ? (Max flows exist due to linkedness results of Chekuri-Khanna-Shepherd)

21. Rao-Zhou Analysis • Partitioning • Randomly partition edges into polylog( | T | ) pieces • Solve max-flow in each piece • How do we know this is feasible? • p-skeletons • (Karger) If min-cut in graph is large, all cuts are preserved in each component up to polylog( | T | ) factors • Can connect “enough” terminals using max-flow-min-cut thm

22. Rao-Zhou Analysis • Build expanders • Use paths created in the previous phase to build expander on the terminals • Route original demands • Now use standard polylog(N)-approx for routing in expanders

23. How to Attack General Case • Obviously wrong approach • Solve EDPwC in expanders • Prove that every graph is an expander

24. How to Attack General Case • Different approach • Prove that every graph is an expander-of-expanders • Solve EDPwC in expanders-of-expanders ? • Use Räcke decomposition result ?

25. Räcke Decompositions • Introduced for oblivious routing • wl(S) = # edges in S between two level l clusters • (Räcke) Can create log N levels s.t. • for all small S in level l cluster U • cap(S, U - S) ≥ wl +1(S) / log N

26. Räcke Decompositions • Introduced for oblivious routing • wl(S) = # edges in S between two level l clusters • (Räcke) Can create log N levels s.t. • for all small S in level l cluster U • cap(S, U - S) ≥ wl +1(S) / log N • Contract level l+1 clusters • Get graph with good expansion

27. How to Attack General Case • Different approach • Prove that every graph is an expander-of-expanders • Solve EDPwC in expanders-of-expanders • Use Räcke decomposition result ?

28. Expanders-of-Expanders • How to route in expander of expanders • Route recursively using expander routing? • Seems difficult • At each level we can route a 1/log N fraction of demands routed at higher level expander expander

29. Uniform Decomposition • But do we need to recurse across all levels? • Suppose decomposition is uniform • For each cluster U either: • all subclusters S have out(S) ≤ logp N or • all subclusters S have out(S) ≥ logp N small large

30. Uniform Decomposition • Critical clusters • Cluster U is critical if: • U is large • all subclusters S of U are small

31. Three Step Routing • Step 1 • Shrink critical clusters to single nodes • All cuts are now large • Use Rao-Zhou

32. Three Step Routing • Step 2 • Shrink small clusters to single nodes • Critical clusters now look like expanders • Route in critical clusters using expander routing

33. Three Step Routing • Step 3 • Route across small clusters • Small clusters have polylog( N ) terminals • Using Rao-Zhou in small clusters gives poly( log log N ) congestion

34. Non-Uniform Result • What could a non-uniform decomposition look like? • Set of critical clusters joined by trees critical cluster

35. Non-Uniform Result • What could a non-uniform decomposition look like? • Set of critical clusters joined by trees Not an acceptable cluster

36. Non-Uniform Result • What could a non-uniform decomposition look like? • Set of critical clusters joined by trees

37. Non-Uniform Result • What could a non-uniform decomposition look like? • Set of critical clusters joined by trees

38. Non-Uniform Result • How do we deal with nodes outside critical cluster? • Replace each node in tree by an expander

39. Non-Uniform Result • How do we deal with nodes outside critical cluster? • Use expanders to route on tree using low congestion paths

40. Non-Uniform Result • Use paths to join up critical clusters • Key property #1: all tree nodes are gone • Key property #2: all cuts are preserved (more or less) • Use analysis for uniform decompositions!!

41. Non-Uniform Result • But what about general non-uniform decompositions? • H = nodes that aren’t in a critical cluster critical cluster H

42. Non-Uniform Result • But what about general non-uniform decompositions? • H = nodes that aren’t in a critical cluster H

43. Non-Uniform Result • Use Racke lemma to find small clusters around outside of H • Let H’ be the subset of H that is not in any of these clusters • |out( H )| ≤ |out( H’ )| / 2 • Can do this at most log(N) times H H’

44. Non-Uniform Result • Repeat log N times • Use small clusters to grow paths of logarithmic length that connect up critical clusters H

45. Non-Uniform Result • So now we have critical clusters joined by paths of logarithmic length (just as in the “tree” case) • We are done… critical cluster H

46. Thank you