1. Multicommodity flow, well-linked terminals and routing problems Chandra Chekuri Lucent Bell Labs Joint work with Sanjeev Khanna and Bruce Shepherd Mostly based on paper in STOC ‘05

2. Routing Problems 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 Route? EDP: path for each pair, paths edge disjoint NDP: paths are node disjoint AN-Flow: flow of one unit per pair with edge/node capacity equal to 1

3. Disjoint paths vs An-Flow s1s1 s2s2 t1t1 t2t2 s1s1 s2s2 t1t1 t2t2 1/2

4. Setup Terminals: X = {s 1,t 1,s 2,t 2,...,s k,t k } each terminal occurs in exactly one pair, |X| = 2k Pairs: matching M on X Instance: (G,X,M) unit capacity graph Focus: edge problems, EDP and An-flow.

5. Multicommodity Flow Formulation (IP) P (i) : set of paths between s i and t i P = P (1) [ P (2)... [ P (k) f(p) : 1 if flow on path p 2 P, 0 otherwise x i : 1 if s i t i is routed, 0 otherwise max  i x i s.t x i =  p 2 P (i) f(p) 1 · i · k  p: e 2 p f(p) · 1 e 2 E x i, f(p) 2 {0,1}

6. Multicommodity Flow Formulation (LP) P (i) : set of paths between s i and t i P = P (1) [ P (2)... [ P (k) f(p) : flow on path p 2 P x i : amount of flow routed for s i t i max  i x i s.t x i =  p 2 P (i) f(p) 1 · i · k  p: e 2 p f(p) · 1 e 2 E x i, f(p) 2 [0,1]

7. Framework 1. Start with an LP solution. 2. Use LP solution to decompose the input instance into a collection well-linked instances. 3. Use well-linkedness to route large fraction

8. Outline Cut vs Flow well-linkedness Well-linked decomposition Multicommodity flow to well-linked decomp decomposition via cuts fractional well-linkedness to well-linkedness Conclusions

9. Multicommodity Flows MC Flow instance: capacitated graph G non-negative demand matrix d on V x V route d ij flow for node pair ij Product MC Flow instance: node weights  : V ! R + implicitly defines d with d ij =  (i)  (j) /  (V)

10. Sparse Cuts and Multicomm. Flow Given a cut (S, V-S) in G and demand matrix d: sparsity of S = |  (S)| / d(S,V-S) MCflow for d is feasible in G implies sparsity ¸ 1 S V - S

11. Sparse Cuts and MC Flow MCflow for d is feasible in G implies sparsity ¸ 1 d is feasible in G if sparsity =  (log n) [LR88,LLR94,AR94] For product MC Flow in planar G, sparisty of  (1) sufficient [KPR93] Flow-cut gap  (G): minimum sparsity reqd for guaranteeing mc flow

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

13. Flow-Well-linked Set Subset X is flow-well-linked in G if the following multicommodity flow is feasible in G: for u,v in X, d(uv) = 1/|X| product (uniform) multicommodity flow on X  (u) = 1 if u 2 X = 0 otherwise

14. Cut vs Flow well-linked X flow-linked ) X is ~cut-linked X cut-linked ) X flow-linked with congestion  (G)  (G) – worst case flow-cut gap for product multicommodity instances in G

15. Weighted versions  : X ! R + weight function on X  (v) : weight of v in X  -cut-linked: for all S ½ V with  (S Å X) ·  (X)/2, |  (S)| ¸  (S Å X)  -flow-linked: multicommodity flow instance with d(uv) =  (u)  (v) /  (X) is feasible in G

16. Well-linked instance of EDP Input instance: G, X, M X = {s 1, t 1, s 2, t 2,..., s k, t k } – terminal set M : matching on X (s 1,t 1 ), (s 2,t 2 )... (s k,t k ) X is well-linked in G

17. Well-linked instance: weighted Input instance: G, X, M X = {s 1, t 1, s 2, t 2,..., s k, t k } – terminal set M : matching on X (s 1,t 1 ), (s 2,t 2 )... (s k,t k ) X is  -well-linked in G for some  : X ! R + Assume:  (v) · 1

18. Examples s1s1 t1t1 s2s2 t2t2 s3s3 t3t3 s4s4 t4t4 Not a well-linked instance s1s1 t1t1 s2s2 t2t2 s3s3 t3t3 s4s4 t4t4 A well-linked instance

19. Well-linked Decomposition G, X, M G 1, X 1, M 1 M i ½ M X i is well-linked in G i  i |X i | ¸ OPT/  G 2, X 2, M 2 G r, X r, M r edge disjoint subgraphs

20. Example s1s1 t1t1 s2s2 t2t2 s3s3 t3t3 s4s4 t4t4 s1s1 t1t1 s2s2 t2t2 s3s3 t3t3 s4s4 t4t4

21. Well-linked Decomposition via Flow G, X, M Flow f G 1, X 1, M 1 X i is  i -flow-well-linked in G i  i  i (X i ) ¸ f/  G 2, X 2, M 2 G r, X r, M r

22. Decomposition via trees/Racke Simple decomposition for trees:  = O(1) Represent G as a tree (approximately) [Racke03] Done in [CKS04] Decomposition based on recursive cuts [CKS05] simple better ratio applies to node problems

23. Trees Define  : X ! R +  (s j ) =  (t j ) = f j the flow in LP Suppose X is  /10-flow-well-linked done! Otherwise exists cut of sparsity less than 1/10 Pick sparse cut (S,V-S) with S minimal

24. Trees S V - S c e <  (S)/10 terminals in S are  -well-linked!

25. Decomposition using Sparse Cuts Start with LP soln for given instance f j flow for pair s j t j : assume flow decomposition f =  j f j total flow in LP define  : X ! R +  (s j ) =  (t j ) = f j

26. Decomposition Algorithm If X is  / 10  (G) log k-flow-linked STOP Else Find a (approx) sparse cut (S,V-S) wrt  in G Remove flow on edges of  G (S) G 1 = G[S], G 2 = G[V-S] Recurse on G 1, G 2 with remaining flow

27. Analysis Remaining graphs at end of recursion (G 1,X 1,  1 ), (G 2,X 2,  2 ),...., (G h, X h,  h )  i is the remaining flow for X i X i is  i /10  (G) log k flow-linked in G_i  i  i (X i ) ¸ Original flow - # edges cut

28. Bounding the number of edges cut X is not  / 10  (G) log k flow-linked ) |  G (S)| ·  (S) / 10 log k S V - S

29. Analysis cont Theorem: total number of edge cut is · f/2 T(x): max # of edges cut if started with flow x T(f) · T(f 1 ) + T(f 2 ) + f 1 / 10 log k For f · k, T(f) · f/2

30. Analysis contd  i  i (X i ) ¸ f/2 X i is  i /10  (G) log k flow-well-linked

31. Fractional to integer well-linked Theorem: G, X, M input instance. X is  -flow-well-linked. Then G, X’, M’ s.t M’ ½ M, X’ is flow-well-linked |X’| =  (  (X))

32. Edge case: spanning tree clustering T spanning tree of G, rooted at r T v : subtree rooted at v Can assume maximum degree of T is 4 1. Find deepest node u s.t  (T u ) ¸ 1 Note:  (T u ) · 5 2. Remove T u from T 3. Continue until  (T) · 1

33. Spanning tree clustering 0.50.70.1 0.3 0.4 0.2 0.3 0.80.4 0.6

34. Spanning tree clustering 0.50.70.1 0.3 0.4 0.2 0.3 0.80.4 0.6 0.50.7 0.2 0.3 0.80.4 0.1 0.3 0.4 0.6

35. Tree clustering T 1, T 2,..., T h clusters Claim: h =  (  (X)) Y is a set of representatives if Y Å T i · 1 for all i Lemma: Y is ½ - flow-well-linked

36. Representatives are well-linked 0.50.7 0.2 0.3 0.1 0.3 0.4 0.6 0.80.4

37. Representatives Need representatives Y such that Y ½ X i Y induces a large submatching of M i Simple greedy scheme works pick s i and t i remove all terminals in trees of s i and t i continue

38. Node case Well-linked decomposition same as for edge case Use node-separators instead of edge separators Clustering is not straighforward (can’t assume degree bound) In [CKS05] weaker bounds than for edge case Recent work: same as for edge case. More technically involved

39. Lower Bounds Well-linked decomposition has to lose  (log 1/2 n) factor Implicitly from integrality gap results for all-or- nothing flow problem [Chuzhoy-Khanna05] Conjecture:  (log n) factor lower bound

40. Flows, Cuts, and Integer Flows max integer flow max frac flow min multicut · · NP-hard Solvable Flow-cut gap thms [LR88...] ?? + graph theory

41. Weaker decomp for planar graphs Well-linked decomp yields O(log n) approx for planar graph EDP (congestion 2) Recent result for planar EDP: O(1) approx with congestion 4 [CKS 05] Weaker decomp based on planar graph properties. Q: well-linked in planar loses  (log n) ?

42. Open problems Improve upper/lower bounds on well-linked decomposition.  (log n)? Approx algorithms for EDP/NDP in general graphs with congestion O(1) essentially reduced to a graph theory problem Directed graphs?

43. Thank You!

44. Trees to Graphs using Racke Hierarchical graph decomposition [Racke03] Given graph G, exists capacitated tree T(G) s.t T(G) approximates G w.r.t sparse cuts Approximation factor – O(  (G) log n log log n) [Harrelson-Hildrum-Rao04] Apply algo. on T(G) to get decomposition for G Loss: polylog(n)