Julia Chuzhoy (TTI-C) Yury Makarychev (TTI-C) Aravindan Vijayaraghavan (Princeton) Yuan Zhou (CMU)

1. Approximating k-route cuts Julia Chuzhoy (TTI-C) Yury Makarychev (TTI-C) Aravindan Vijayaraghavan (Princeton) Yuan Zhou (CMU)

2. Cut minimization • Min st-cut: delete the min #edges to disconnect s, t t Duality: Maxflow(s, t) = Mincut(s, t) s = 2

3. Multicut • Given r pairs (si, ti), delete min #edges to disconnect all (si, ti) pairs t1 • Upper bound on max multicommodity flow • Identifies bottlenecks in the graph • O(log r) approximation algorithm [GVY95] t3 s3 s2 t2 s1

4. Min k-route cuts • Unweighted version. Given r pairs (si, ti), delete min #edges to k-disconnect all (si, ti) pairs • i.e. for all i, (si, ti)-edge-connectivity < k • General version. Given a weighted graph and r pairs, delete min wt. of edges to k-disconnect all (si, ti) pairs t1 For example, when k = 2, OPT = 1. t3 s3 s2 t2 s1

5. Min k-route cuts: variants and specal cases • EC-kRC: edge connectivity version, remove min. wt. of edges so that for each i, (si, ti)-edge-connectivity < k • Unweighted case: all edge weights = 1 • k = 1: Minimum multicut • s-t EC-kRC: single source-sink pair version • VC-kRC: vertex connectivity version, remove min. wt. of edges so that for each i, (si, ti)-vertex-connectivity < k

6. Motivation : a fault tolerant setting • Multiroute generalization • st-k-route flow: a fractional combination of elementary k-route st-flows [Kis96, KT93, AO02] • Flow is resilient to (k-1) failures multiroute generalization Maxflow/ Mincut st-k-route flow multicommodity flow k-route multicommodity flow multicut k-route cut

7. Motivation (cont'd) • Multiroute generalization: a fault tolerant setting • As standard multicut, k-route cut also reveals network bottleneck, and in particular measures resilience of the network multicut k-route cut

8. Approximation algorithms • α-approximation: delete edges of wt. αOPT such that all the pairs are k-disconnected • (β,α)-bicriteria approximation: delete edges of wt. αOPT such that all the pairs are βk-disconnected

9. Previous work • [Chekuri-Khanna'08] • O(log2n log r)-approximation for k=2 (both EC-2RC and VC-2RC) • [Barman-Chawla'10] • O(log2r)-approximation for k=2 (both EC-2RC and VC-2RC) • NP-Hardness for s-t EC-kRC • [Kolman-Scheideler'11] • O(log3r)-approximation for k=3 (EC-2RC) • No sub-polynomial approx. algorithm known for k > 3

10. Our results : algorithms for EC-kRC • Unweighted EC-kRC • O(k log1.5 r)-approximation • (1+ε, (1/ε)log1.5 r)-bicriteria approximation • General EC-kRC • O(log1.5 r)-approximation for k = 2 • (2, log2.5 r loglog r)-bicriteria approx. in nO(k) time • (log r, log3 r)-bicriteria approx. in poly(n, k) time

11. Our results : VC-kRC • Algorithms • O(log1.5 r)-approximation for k = 2 • (2, d k log2.5 r loglog r)-bicriteria approx. in nO(k) time, where each node belongs to at most d source-sink pairs • Harndess for VC-kRC • NP-Hard to approximate VC-kRC within Ω(kε) for some specific ε > 0 • Hardness for st-VC-kRC • Superconstant hardness assuming random k-AND hypothesis of [Feige'02] • Ω(ρ0.5) hardness assuming ρ-inapproximability of Densest k-Subgraph

12. A comparison : EC-kRC

13. A comparison : VC-kRC

14. Why only bicriteria algorithm for large k? • The problem might be hard even for single s-t version • st-VC-kRC: no sub-polynomial approx. if assuming no sub-polynomail approx. for Densest k-Subgraph • st-EC-kRC: no sub-polynomial approx. known (there is a (2, 2)-bicriteria approx. alg.) • Embarrassing situation for st-EC-kRC: even APX-hardness is not known • Recall the problem: • A weighted graph G, • source s, sink t • Goal: remove k edges in G to minimize min s-t cut

15. The rest of this talk... • O(k log1.5 r)-approximation algorithm for unweighted EC-kRC • O(log1.5 r)-approximation algorithm for general EC-2RC • (2, log2.5 r loglog r)-bicriteria approx. algorithm for general EC-kRC (sketch)

16. The difficulty for large k (> 2) • Simple recursion (used in [BC10]) for k = 2 • Find a balanced cut (by region growing) • Remove all the cut edges but the most expensive one • recurse into both sides • Key observation. the red edge cannot provide extra connectivity for s1, t1 graph G s1 t1

17. The difficulty for large k (> 2) • Simple recursion (used in [BC10]) for k = 2 • Find a balanced cut (by region growing) • Remove all the cut edges but the most expensive one • recurse into both sides • Key observation. the red edge cannot provide extra connectivity for s1, t1 • No longer true for k = 3 (or more) graph G s1 t1 a bad example for k = 3

18. Algorithms for k > 2 • [Kolman-Scheideler'11] O(log3r)-approximation for k=3, by multi-level region growing (based on the same LP used in [BC10]) • Our method • Idea 1. Relate k-route cut to the value of sparest cut • Idea 2. Solve the problem iteratively rather than recursively

19. O(k log1.5 r)-approximation algorithm for unweighted EC-kRC

20. Cut sparsity, and unweighted EC-kRC • Let d(v) = #source-sink pairs that v participates in d(S) = • Define uniform sparsity to be • Intuition. Given a cut , is small when • the cut size is small • the cut separates many terminals (or, the cut is balanced in terms of d)

21. Lemma. Cut sparsity, and unweighted EC-kRC • Let d(v) = #source-sink pairs that v participates in d(S) = • Define uniform sparsity to be • Theorem.[ARV04]O(log0.5 r)-approx. for Φ(G).

22. Lemma. Algorithm for unweighted EC-kRC • Step 0. Assume source-sink pairs are not k-disconnected • Step 1. Use the algorithm in [ARV04] to find an approximate sparse cut • Step 2. Delete all the edges across the cut • Step 3. Recurse into the subinstances defined by each side of the cut • Fact. #cut edges deleted in Step 2 is at most

23. A standard charging argument • Fact. #cut edges deleted in Step 2 is at most • Charge this cost to the smaller part among • At one step, each terminal is charged by • Each terminal can be in "small parts" • In total, each terminal is charged by • Since there are r terminals, total cost:

24. Lemma. Proof of • Consider H = G \ OPT • For every (si, ti) pair, mincutH(si, ti) = |edges(Si, Ti)| < k (a witness cut) • Claim. The witness cuts are laminar Si si ti Ti

25. Proof of Claim: witness cuts are laminar • Gomory-Hu Tree. (exists for every graph) A weighted tree that consists of edges representing all pairs minimum s-t cuts in the graph. mincutH(s, t) = mincutT(s, t) • All s-t mincuts in the tree are laminar ==> All mincuts in H are laminar ==> All witness cuts are laminar H: Gomory-Hu tree T

26. Lemma. Proof of • Consider H = G \ OPT • For every (si, ti) pair, mincutH(si, ti) = |edges(Si, Ti)| < k (a witness cut) • Claim. The witness cuts are laminar • Let S1, S2, ..., Sm be the maximal witness cuts (the smaller parts) S2 S1 S3

27. Lemma. Proof of • Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT 1. d(S1) + d(S2) + ... + d(Sm) >= r 2. therefore S2 S1 S3

28. Lemma. Proof of • Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT 1. d(S1) + d(S2) + ... + d(Sm) >= r 2. S2 S1 (since each edge is shared by at most 2 maximal cuts) S3

29. Lemma. Proof of • Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT 1. d(S1) + d(S2) + ... + d(Sm) >= r 2. 3. by expansion In all: S2 S1 S3

30. O(log1.5 r)-approx. algorithm for EC-2RC

31. Another definition of sparsity the most expensive edge • 2-route uniform sparsity • Corollary of[ARV04]. 2-route uniform sparisty can be efficiently approximated within O(log0.5 r) factor • Proof. Guess the red edge, remove it, and run ARV. S S

32. Lemma. Another definition of sparsity the most expensive edge • 2-route uniform sparsity • Corollary of[ARV04]. 2-route uniform sparisty can be efficiently approximated within O(log0.5 r) factor S S

33. Algorithm for EC-2RC • Step 1. Find an approximate 2-route sparse cut • Step 2. Delete all but the most expensive edge across the cut • Step 3. Recurse into the subinstances defined by each side of the cut • Claim. The algorithm outputs a valid 2-route cut. • Proof. By the key observation we made before.

34. Lemma. Algorithm for EC-2RC • Step 1. Find an approximate 2-route sparse cut • Step 2. Delete all but the most expensive edge across the cut • Step 3. Recurse into the subinstances defined by each side of the cut • Fact. wt of edges deleted in Step 2 is at most • Corollary. wt of edges removed in total

35. Lemma. Proof of • Consider H = G \ OPT • For every (si, ti) pair, mincutH(si, ti) = |edges(Si, Ti)| < 2 (a witness cut) • Claim. The witness cuts are laminar • Let S1, S2, ..., Sm be the maximal witness cuts (the smaller parts) S2 S1 S3

36. Lemma. Proof of • Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT 1. d(S1) + d(S2) + ... + d(Sm) >= r 2. S2 S1 , therefore, S3

37. Lemma. Proof of • Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT 1. d(S1) + d(S2) + ... + d(Sm) >= r 2. In all, thus, there exists i:

38. (2, log2.5 r loglog r)-bicriteria approx. for general EC-kRC (sketch)

39. Lemma. (2, log2.5 r loglog r)-bicriteria approx. for general EC-kRC (sketch) • k-route non-uniform sparsity where • Corollary. (of [ALN05]) O(log0.5 r loglog r) approx. in nO(k) time : total wt of all the edges across the cut but the most expensive (k-1) ones : #source-sink pairs across the cut

40. Lemma. (2, log2.5 r loglog r)-bicriteria approx. for general EC-kRC (sketch) (cont'd) • The iterative algorithm. (Applying Idea 2) • Step 1. Use the algorithm in [ALN05] to find an approximate sparse cut • Step 2. Delete all the edges across the cut but the (2k-2) most expensive ones • Step 3. Remove all the source-sink pairs that are (2k-1)-disconnected • Step 4. Repeat Step 1~3 until no source-sink pair remains • Theorem. Wt. of removed edges <= log2.5 r loglog r OPT

41. Open questions • Algorithm side. • Better true approximation algorithm for general EC-kRC (and VC-kRC) • Hardness side. • Is EC-kRC (for large k) strictly harder than multicut? • Understand the simplest case: st-EC-kRC.

42. Thank you!