1. IMIM v 11 00000 v 2 1 10000 v 31 00000 v 12 01000 v 2 2 00100 v 32 00010 v 13 00000 v 23 00001 v 33 00000 DEFINITION L v 11 v 2 1 v 31 v 12 v 2 2 v 32 v 13 V 23 v 33 a1a1 111000000 b1b1 000111000 c1c1 000000111 Undirected GraphsDirected Graphs O(min(log n, log k)) [Leighton, Rao ’88, Garg, Vazirani, Yannakakis ’93] APX-hard [Chuzhoy, Khanna ’09] O(n 11/23 ) - approx alg [Agarwal, Alon, Charikar ’07] 2 Ω( log 1- ε n ) hardness [Chuzhoy, Khanna ’09] The State of the Art s s s t t t The Maximum Multicommodity Flow Problem s s s t t t 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 Find: A maximum total weight set of fractional s i -t i paths. Problem: Large integrality gap Undirected: equal to best multicut approx Directed: Ω(min(( k, n δ )) [Saks, Samorodnitsky, Zosin ’04, Chuzhoy, Khanna ’09 ] min Σ e in E x e such that Σ e in p x e ≥ 1 1 ≤ i ≤ k, p s i -t i path x e ≥ 0 e ∈ E All approximation algorithms for multicut use LP relaxation with maximum multicommodity flow problem as dual. DEFINITION Hypergrid( n, k ) is the k -fold strong product of P n. It has n k nodes and k s-t pairs. DEFINITION A network is: Undirected graph G = (V,E ), nodes capacity one Subsets of V : {S i } i ∈[k], {T i } i ∈[k] s i - t i pairs, i ∈[k], connect to G: s i connects to v ∈S i, and v ∈T i connect to t i DEFINITION The strong product of two networks N and N’, denoted N ☒ N’, is network: G = ( V×V’, {((u,u’), (v,v’)) |(u,v)∈E or u=v, (u’,v’)∈E’ or u’=v’}) Source subsets: {S i × V’} i ∈[k] U {V × S’ i } i ∈[k’] Sink subsets: {T i × V’} i ∈[k] U {V × T’ i } i∈[k’] The n-path network P n = Hypergrid (n,1) : v 1 v 2 v 3 v 4 v 5 v n-1 v n s1s1 t1t1 Hypergrid( 3, 2 ) DEFINITION A coding matrix is decodable with rate p if ∃ subset D of messages, |D| = p, such that: For all i, for all message m i ∈ D originating at s i, m i is a linear combination of columns in T i. A coding matrix describes a linear code of N if: Column v describes the linear combination of messages sent by v Column v is a linear combination of columns of predecessors of v. Hypergrid( 3, 3 ) ( v 1,v 1 ’ )( v 2,v 1 ’ )( v 3,v 1 ’ )( v 1,v 2 ’ )( v 2,v 2 ’ )( v 3,v 2 ’ )( v 1,v 3 ’ )( v 2,v 3 ’ )( v 3,v 3 ’ ) a1a1 111000000 b1b1 000111000 c1c1 000000111 a2a2 100100100 b2b2 010010010 c2c2 001001001 Coding matrix for Hypergrid( 3, 2 ) S 1 T 1 S 2 T 2 S 3 T 3 The optimal max multicommodity flow solution for Hypergrid( n, k ) routes n k-1 units of flow between a single s-t pair. Hypergrid( 3, 2 ) is decodable with rate 5 using D = {a 1, b 1, c 1, a 2, b 2 }. Proof: M a multicut: at least one non-zero in each row of L I M Disjoint paths: at most one non-zero in each column of L I M Proof: M a multicut: at least one non-zero in each row of L I M Disjoint paths: at most one non-zero in each column of L I M v ij abbreviation for ( v i, v j ’ ) Messages a i, b i, c i originate at s i Matrix I M represents a cut M. OBSERVATION If there is a matrix L s.t. rank (L I M ) ≥ p for all multicuts M, then the minimum multicut is at least p. OBSERVATION If L is the coding matrix for a solution that routes messages along p node-disjoint paths, then rank (L I M ) ≥ p for any multicut M. Proof: For M a minimum multicut, |M | = rank ( I M ) ≥ rank (L I M ) ≥ p. Proof: For M a minimum multicut, |M | = rank ( I M ) ≥ rank (L I M ) ≥ p. Is network coding a better lower bound on multicut than 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 + edges between s i and t j for all i ≠ j Flow Rate = Multicut = 1, Coding Rate = k THEOREM Given node-capacitated networks N 1 and N 2 with coding matrices L 1 and L 2, there is a coding matrix for N 1 ☒ N 2 : L = such that: 1.If L 1 and L 2 are decodable with rates p 1 and p 2 then L is decodable with rate p := n 1 p 2 + n 2 p 1 - p 1 p 2. 2.If L 1 and L 2 are p 1 and p 2 certifiable then L is p- certifiable. DEFINITION A coding matrix L is p -certifiable if 1.Column v of L is a linear combination of columns of incoming sources and predecessors of v that form a clique. 2.For any multicut M, rank (L I M ) ≥ p. OBSERVATION A code that routes messages along p node- disjoint paths is p -certifiable. KEY LEMMA For all multicuts M of N 1 ☒ N 2 and cliques K in V 1 (V 2 ) there is a multicut M K of N 2 (N 1 ) such that K ⊗ M K (M K ⊗ K) is a subset of M. Example: Let K = {v 1, v 2 }. The copies of N 2 corresponding to v 1 (yellow) and v 2 (blue) must contain the same multicut. Proof of 1 A direct consequence of definitions. Proof of 1 A direct consequence of definitions. COROLLARY Hypergrid(n,k ) has a code that is decodable with rate n k - (n-1) k and is n k - (n-1) k certifiable. Saks et al. showed multicut is at least k(n-1) k-1 The trivial multicut that cuts all nodes adjacent to sinks is optimal. Proof of 2: Analyze rank ( LB ) for some matrix B in the span of I M. Proof of 2: Analyze rank ( LB ) for some matrix B in the span of I M. K used to decode  certain off diagonal blocks of LB vanish. v1v1 v2v2 vn1vn1 v3v3 … v’ 1 v’ n 2 … v’ 1 v’ n 2 … v’ 1 v’ n 2 … v’ 1 v’ n 2 … 0 0 IMKIMK in K in V 1 rank (L 2 I M K )=p 2 and rank (L 1 I M K’ )=p 1  diagonal blocks have ranks p 1 and p 2 rank (L 2 I M K )=p 2 and rank (L 1 I M K’ )=p 1  diagonal blocks have ranks p 1 and p 2 Hypergrid (n,1) is 1- certifiable. In1⊗ L2In1⊗ L2 L1⊗ In2L1⊗ In2 B has a set of columns for each v ∈ V 1 U V 2 determined by the clique K used to decode v. v’ 1 v’ n 2 … v4v4 … Main proof ideas: 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? capacity 1 All other edges have infinite capacity s 2 t2t2 s1s1 ( v 2, v 1 ’ ) ( v 1, v 2 ’ )( v 3, v 2 ’ )( v 2, v 2 ’ ) ( v 2, v 3 ’ ) t1t1 ( v 1, v 1 ’ )( v 3, v 1 ’ ) ( v 1, v 3 ’ )( v 3, v 3 ’ ) s 2 t2t2 s1s1 ( v 2, v 1 ’ ) ( v 1, v 2 ’ )( v 3, v 2 ’ ) ( v 2, v 2 ’ ) ( v 2, v 3 ’ ) t1t1 ( v 1, v 1 ’ )( v 3, v 1 ’ ) ( v 1, v 3 ’ )( v 3, v 3 ’ ) s 2 t2t2 s1s1 ( v 2, v 1 ’ ) ( v 1, v 2 ’ )( v 3, v 2 ’ )( v 2, v 2 ’ ) ( v 2, v 3 ’ ) t1t1 ( v 1, v 1 ’ )( v 3, v 1 ’ ) ( v 1, v 3 ’ )( v 3, v 3 ’ ) s 2 t2t2 s1s1 ( v 2, v 1 ’ ) ( v 1, v 2 ’ ) ( v 3, v 2 ’ ) ( v 2, v 2 ’ ) ( v 2, v 3 ’ ) t1t1 ( v 1, v 1 ’ ) ( v 3, v 1 ’ ) ( v 1, v 3 ’ ) ( v 3, v 3 ’ ) Key a 1 +a 2 b 1 +a 2 b1b1 a 1 +b 2 a1a1 c1c1 b 1 +b 2 c 1 +a 2 c 1 +b 2 IMKIMK IMKIMK Node capacitated ⊆ Edge capacitated, directed 1 1 1 1 1 1