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

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

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

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

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 VC-kRC: vertex connectivity version, remove min. wt. of edges so that for each i, (si, ti)-vertex-connectivity < k

multiroute generalization 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 : a fault tolerant setting multiroute generalization Maxflow/ Mincut st-k-route flow multicommodity flow k-route multicommodity flow multicut k-route cut

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

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

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 [Kolman-Scheideler'11] O(log3r)-approximation for k=3 (EC-2RC) No sub-polynomial approx. algorithm known for k > 3

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

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

A comparison : EC-kRC Previous results Our results k = 2 O(log2 r) [BC10] O(log1.5 r) k = 3 O(log3 r) [KS11] arbitrary k, unweighted O(k log1.5 r) (1+ε, (1/ε)log1.5 r) general (2, log2.5 r loglog r) in time nO(k) (log r, log3 r) in poly(n, k) time

superconstant assuming UGC A comparison : VC-kRC Previous results Our results k = 2 O(log2 r) [BC10] O(log1.5 r) arbitrary k multicut hardness: APX-hard [DJP+94] superconstant assuming UGC [KV05, CKK+06] (2, dklog2.5r loglog r) alg in time nO(k) Ω(kε)-hardness

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

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

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

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

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

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

Proof of Lemma. 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

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

Proof of Lemma. 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 S2 S1 S3

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

(since each edge is shared by at most 2 maximal cuts) 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

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

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

(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 Lemma.

(2, log2. 5 r loglog r)-bicriteria approx (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 Lemma.

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.

Thank you!