Multicut Lower Bounds via Network Coding

Multicut Lower Boundsvia Network Coding Anna Blasiak Cornell University

Multicut • Given: • Directed graph G = (V,E) • Capacities on edges • k source-sink pairs • Find: • A min-cost subset of E such that on removal all source-sink pairs are disconnected s3 s2 s1 t1 t3 t2

Directed MulticutThe State of the Art Õ(n11/23) - approxalgorithm [Agarwal, Alon, Charikar’07] 2Ω(log1-ε n) hardness [Chuzhoy, Khanna ‘09] Nothing non-trivial known in terms of k

Dual Problem:Maximum Multicommodity Flow • Given: • A directed graph G = (V,E) • Capacities on nodes • k source-sink pairs • Find: • A maximum total weight set of fractional si-ti paths. All approximation algorithms for multicutare based on the LP and compare to the maximum multicommodityflow. Limited by large integrality gap: Ω(min((k, nδ)) [Saks, Samorodnitsky,Zosin ’04, Chuzhoy, Khanna’09 ] s3 s2 s1 t1 t3 t2

Better Lower Bound?Network Coding • Good News: Coding Rate ≥ Flow Rate, can be a factor klarger • Bad News: Multicut≱ Coding Rate, can be a factor k smaller

Results • Identify a property Pof a network code such that any code satisfying P is a lower bound on the multicut. • Show Pis preserved under a graph product. • Main Corollary: • Improved and tight lower bound on the multicut in the construction of Saks et al. and give a network code with rate = min multicut

Simplifying Assumptions Type of networks giving: Ω(min((k, nδ)) [Saks, Samorodnitsky,Zosin ’04, Chuzhoy, Khanna’09 ] • A network is: • Undirected graph G = (V,E ) • Capacity one for each node • Subsets of V: {Si}i∈[k], {Ti}i∈[k] si - tipairs, i∈[k], connect to G: • siconnects to v ∈Si • v ∈Ti connect to ti

Saks et al. Construction • Begin with The n-path network Pn: S1 T1 Hypergrid(n, k)is the k-fold strong product of Pn. It has nknodes and k s-t pairs and flow rate nk-1 Saks et al. show it has multicut at least has knk-1 Theorem Hypergrid(n,k) has a code with rate nk- (n-1)kthat is a lower bound on the multicut. s1 t1 v1 v2 v3 v4 v5 vn-1 vn

Hypergrid(3,2) = P3☒P3 s2 (v1 , v1’ ) (v2 , v1’ ) (v3, v1’ ) (v1 , v2’ ) (v2 , v2’ ) (v3, v2’ ) t1 s1 (v3, v3’ ) (v1 , v3’ ) (v2 , v3’ ) t2

Hypergrid(3,3) = P3 ☒P3☒P3 S1T1S2T2S3T3

Coding Matrix Columns labeled with v in V a2, b2, c2 a1+a2 a1+b2 a1 b1+a2 b1+b2 b1 a1, b1, c1 c1 c1+b2 c1+a2 Entries in finite field • Column v describes the linear combination of messages sent by v. • Column v is a linear combination of columns of predecessors of v. • tican decode messages from si. • Rows labeled with messages • ai, bi , cioriginate at si • rate of code = # of messages s2 (v1 , v1’ ) (v2 , v1’ ) (v3, v1’ ) (v3, v2’ ) (v1 , v2’ ) (v2 , v2’ ) t1 s1 (v3, v3’ ) (v1 , v3’ ) (v2 , v3’ ) t2

Coding Matrix as a Lower Bound definition |V|x|M| matrix, column v ∈ M is an indicator vector for v. A coding matrix L is p-certifiable if For any multicutM,rank(LIM)≥ p. Column vof L is a linear combination of columns of incoming sources and predecessors of vthat form a clique. L gives a lower bound on the multicut: For M a minimum multicut, |M| = rank (IM) ≥ rank (LIM) ≥ p.

Main Theorem* Given networks N1and N2 with coding matrices L1and L2, there is a coding matrix for N1☒ N2 : L =such that: If L1and L2 have rates p1and p2 then L has rate p := n1p2 + n2 p1 - p1p2. If L1and L2 are p1and p2 certifiable then L is p-certifiable.

Saks et al. Construction • Begin with The n-path network Pn: Hypergrid(n, k)is the k-fold strong product of Pn. It has nknodes and k s-t pairs and flow rate nk-1 Saks et al. show it has multicut at least has knk-1 Pnis 1-certifiable. Theorem Hypergrid(n,k) has a code with rate nk-(n-1)kthat is a lower bound on the multicut. s1 t1 v1 v2 v3 v4 v5 vn-1 vn

Conclusions and Open Questions • Our result: a certain type of network coding solution is a lower bound on directed multicut • Is there a more general class of network coding solutions that is a lower bound? • Multicommodity flow can be far from the multicut, what about the network coding rate? • Does there exist an α = o(k) s.t.multicut ≤ α network coding rate?

Multicut Lower Bounds via Network Coding