1. 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 t s s Duality: Maxflow(s, t) = Mincut(s, t) = 2

3. Multicut Given r pairs (s i, t i ), delete min #edges to disconnect all (s i, t i ) pairs t1t1 t1t1 s1s1 s1s1 s2s2 s2s2 s3s3 s3s3 t3t3 t3t3 Upper bound on max multicommodity flow Identifies bottlenecks in the graph O(log r) approximation algorithm [GVY95] t2t2 t2t2

4. Min k-route cuts Unweighted version. Given r pairs (s i, t i ), delete min #edges to k-disconnect all (s i, t i ) pairs –i.e. for all i, (s i, t i )-edge-connectivity < k General version. Given a weighted graph and r pairs, delete min wt. of edges to k-disconnect all (s i, t i ) pairs For example, when k = 2, OPT = 1. t1t1 t1t1 s1s1 s1s1 s2s2 s2s2 s3s3 s3s3 t3t3 t3t3 t2t2 t2t2

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

6. Motivation 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 Maxflow/ Mincut multiroute generalization st-k-route flow multicutk-route cut multicommodity flow k-route multicommodity flow : a fault tolerant setting

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 multicutk-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(log 2 n log r)-approximation for k=2 (both EC-2RC and VC-2RC) [Barman-Chawla'10] –O(log 2 r)-approximation for k=2 (both EC-2RC and VC-2RC) [Kolman-Scheideler'11] –O(log 3 r)-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 log 1.5 r)-approximation –(1+ε, (1/ε)log 1.5 r)-bicriteria approximation General EC-kRC –O(log 1.5 r)-approximation for k = 2 –(2, log 2.5 r loglog r)-bicriteria approx. in n O(k) time –(log r, log 3 r)-bicriteria approx. in poly(n, k) time

11. Our results : VC-kRC Algorithms –O(log 1.5 r)-approximation for k = 2 –(2, d k log 2.5 r loglog r)-bicriteria approx. in n O(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(k ε ) for some specific ε > 0 Hardness for st-VC-kRC –Superconstant hardness assuming random k-AND hypothesis of [Feige'02] –Ω(ρ 0.5 ) hardness assuming ρ-inapproximability of Densest k-Subgraph

12. A comparison : EC-kRC Previous resultsOur results k = 2O(log 2 r) [BC10]O(log 1.5 r) k = 3O(log 3 r) [KS11] arbitrary k, unweighted O(k log 1.5 r) (1+ε, (1/ε)log 1.5 r) arbitrary k, general (2, log 2.5 r loglog r) in time n O(k) (log r, log 3 r) in poly(n, k) time

13. A comparison : VC-kRC Previous resultsOur results k = 2O(log 2 r) [BC10]O(log 1.5 r) arbitrary k multicut hardness: APX-hard [DJP + 94] superconstant assuming UGC [KV05, CKK + 06] (2, dklog 2.5 r loglog r) alg in time n O(k) Ω(k ε )-hardness

14. The rest of this talk... O(k log 1.5 r)-approximation algorithm for unweighted EC- kRC (2, log 2.5 r loglog r)-bicriteria approx. algorithm for general EC-kRC (sketch)

15. 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 s 1, t 1 graph G s1s1 t1t1

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 s 1, t 1 No longer true for k = 3 (or more) graph G s1s1 t1t1 a bad example for k = 3

17. Algorithms for k > 2 [Kolman-Scheideler'11] O(log 3 r)-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

18. O(k log 1.5 r)-approximation algorithm for unweighted EC-kRC

19. 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(log 0.5 r)-approx. for Φ(G). Lemma.

20. 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 Corollary. #edges deleted in total is at most Lemma.

21. Proof of Consider H = G \ OPT For every (s i, t i ) pair, mincut H (s i, t i ) = |edges(S i, T i )| < k (a witness cut) Claim. The witness cuts are Laminar Lemma. sisi titi SiSi TiTi

22. 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. mincut H (s, t) = mincut T (s, t) All s-t mincuts in the tree are Laminar ==> All mincuts in H are Laminar ==> All witness cuts are Laminar H:H: H:H: Gomory-Hu tree T

23. Proof of Consider H = G \ OPT For every (s i, t i ) pair, mincut H (s i, t i ) = |edges(S i, T i )| < k (a witness cut) Claim. The witness cuts are Laminar Let S 1, S 2,..., S m be the maximal witness cuts Lemma. S1S1 S2S2 S3S3

24. Proof of Let S 1, S 2,..., S m be the maximal witness cuts in H=G\OPT 1. d(S 1 ) + d(S 2 ) +... + d(S m ) >= r 2. therefore Lemma. S1S1 S2S2 S3S3

25. Proof of Let S 1, S 2,..., S m be the maximal witness cuts in H=G\OPT 1. d(S 1 ) + d(S 2 ) +... + d(S m ) >= r 2. Lemma. S1S1 S2S2 S3S3 (since each edge is shared by at most 2 maximal cuts)

26. Proof of Let S 1, S 2,..., S m be the maximal witness cuts in H=G\OPT 1. d(S 1 ) + d(S 2 ) +... + d(S m ) >= r 2. 3. by expansion In all: Lemma. S1S1 S2S2 S3S3

27. (2, log 2.5 r loglog r)-bicriteria approx. for general EC-kRC (sketch)

28. k-route non-uniform sparsity where Corollary. (of [ALN05]) O(log 0.5 r loglog r) approx. in n O(k) time : total wt of all the edges across the cut but the most expensive (k-1) ones : #source-sink pairs across the cut Lemma.

29. (2, log 2.5 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 <= log 2.5 r loglog r OPT Lemma.

30. 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.

31. Thank you!