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1. Multicut lower bounds via network coding Anna Blasiak Cornell University Multicut Results • Given: • A graph G = (V,E) • Capacities for each edge • k source-sink pairs • Find: • A min-cost subset of E such that on removal all source-sink pairs are disconnected The n-path network Pn= Hypergrid(n,1): s s s s s s Node capacitated ⊆ Edge capacitated, directed 1 1 Key 1 S1T1S2T2S3T3 1 Hypergrid(3, 3) Hypergrid(3, 2) s2 s2 s2 Flow Rate = Multicut = 1, Coding Rate = k The optimal max multicommodity flow solution for Hypergrid(n, k)routes nk-1units of flow between a single s-t pair. The trivial multicut that cuts all nodes adjacent to sinks is optimal. s2 1 1 Theorem definition observation observation Observation definition definition Definition Definition definition Key lemma t t Corollary The State of the Art t t t t (v1 , v1’ ) (v1 , v1’ ) (v1 , v1’ ) (v1 , v1’ ) (v2 , v1’ ) (v2 , v1’ ) (v2 , v1’ ) (v2 , v1’ ) (v3, v1’ ) (v3, v1’ ) (v3, v1’ ) (v3, v1’ ) Given node-capacitated networks N1and N2 with coding matrices L1and L2, there is a coding matrix for N1☒ N2 : L =such that: If L1and L2 are decodable with rates p1and p2 then L is decodable with rate p := n1p2 + n2 p1- p1p2. If L1and L2 are p1and p2 certifiable then Lis p-certifiable. For all multicutsMof N1☒N2 and cliques Kin V1 (V2)there is a multicutMK of N2(N1)such that K⊗MK(MK⊗K)is a subset of M. • The strong product of two networks N and N’, denoted N ☒N’, is network: • G = ( V×V’, {((u,u’), (v,v’)) |(u,v)∈E oru=v, (u’,v’)∈E’oru’=v’}) • Source subsets: {Si × V’}i∈[k]U{V × S’i}i∈[k’] • Sink subsets: {Ti × V’}i∈[k]U{V × T’i}i∈[k’] A coding matrix is decodable with rate pif∃subsetDof messages, |D|= p, such that: For all i, for all message mi∈ D originating at si, miis a linear combination of columns in Ti. • A network is: • Undirected graph G = (V,E ), nodes capacity one • Subsets of V: {Si}i∈[k], {Ti}i∈[k] • si - tipairs, i∈[k], connect to G: • siconnects to v ∈Si , and v ∈Ti connect to ti Hypergrid(n, k)is the k-fold strong product of Pn. It has nknodes and k s-t pairs. A coding matrix L is p-certifiable if Column vof L is a linear combination of columns of incoming sources and predecessors of vthat form a clique. For any multicutM,rank(LIM)≥ p. Hypergrid(n,k) has a code that is decodable with rate nk- (n-1)kand is nk- (n-1)k certifiable. t1 t1 t1 t1 s1 s1 s1 s1 (v2 , v2’ ) (v1 , v2’ ) (v1 , v2’ ) (v1 , v2’ ) (v1 , v2’ ) (v2 , v2’ ) (v2 , v2’ ) (v2 , v2’ ) (v3, v2’ ) (v3, v2’ ) (v3, v2’ ) (v3, v2’ ) s1 t1 v1 v2 v3 v4 v5 vn-1 vn (v1 , v3’ ) (v1 , v3’ ) (v1 , v3’ ) (v1 , v3’ ) (v2 , v3’ ) (v2 , v3’ ) (v2 , v3’ ) (v2 , v3’ ) (v3, v3’ ) (v3, v3’ ) (v3, v3’ ) (v3, v3’ ) Proof of 1 A direct consequence of definitions. t2 t2 t2 t2 The Maximum Multicommodity Flow Problem • Find: • A maximum total weight set of fractional si-ti paths. Example: Let K = {v1, v2}. The copies of N2 corresponding to v1 (yellow) and v2 (blue) must contain the same multicut. • All approximation algorithms for multicut use LP relaxation with maximum multicommodity flow problem as dual. Coding Matrices Cut Lower Bounds via Matrices Matrix IM represents a cut M. IM min Σein E xe such that Σein p xe≥ 1 1 ≤i≤ k, p si-tipath xe≥ 0 e ∈ E L • A coding matrix describes a linear code of N if: • Column v describes the linear combination of messages sent by v • Column vis a linear combination of columns of predecessors of v. Saks et al. Construction: The Hypergrid Proof of 2: Analyze rank(LB) for some matrix B in the span of IM. vijabbreviation for (vi , vj’ ) IMK If L is the coding matrix for a solution that routes messages along p node-disjoint paths, then rank (LIM) ≥ p for any multicutM. If there is a matrix Ls.t. rank(LIM)≥ p for all multicutsM, then the minimum multicutis at least p. A code that routes messages along p node-disjoint paths is p-certifiable. Problem: Large integrality gap Undirected: equal to best multicutapprox Directed: Ω(min((k, nδ)) [Saks, Samorodnitsky, Zosin ’04, Chuzhoy, Khanna’09 ] Main proof ideas: 0 Messages ai, bi , cioriginate at si Coding matrix for Hypergrid(3, 2) 0 K used to decode certain off diagonal blocks of LB vanish. IMK in K in V1 v4 B has a set of columns for each v ∈ V1U V2determined by the cliqueKused to decode v. Guiding Questions Proof: For M a minimum multicut, |M| = rank (IM) ≥ rank (LIM) ≥ p. rank(L2IMK)=p2 and rank(L1IMK’)=p1diagonal blocks have ranks p1and p2 Is network coding a better lower bound on multicutthan maximum multicommodity flow in directed graphs? Good News: Coding Rate ≥ Flow Rate, can be a factor k larger Bad News: Multicut≱ Coding Rate, can be a factor k smaller … … IMK capacity 1 All other edges have infinite capacity + edges between si and tjfor all i ≠ j Proof: M a multicut: at least one non-zero in each row of LIM Disjoint paths: at most one non-zero in each column of LIM Hypergrid(3, 2)is decodable with rate 5 using D = {a1, b1 , c1 , a2, b2 }. v1 Saks et al. showed multicut is at leastk(n-1)k-1 • New Questions: • When is network coding a lower bound on multicut in directed graphs? • When it is an lower bound, is it a better lower bound than maximum multicommodity flow? Does there exist an α = o(k) s.t.multicut ≤ α network coding rate? v’1 v’1 v’1 v’1 v’1 v2 v’n2 v’n2 v’n2 v’n2 v’n2 … … … … … a1+a2 a1+b2 a1 v3 b1+a2 b1+b2 b1 Hypergrid(n,1) is 1-certifiable. c1+a2 c1+b2 c1 vn1