Multicut Lower Bounds via Network Coding Anna Blasiak Cornell University.

Multicut Lower Boundsvia Network Coding

Anna BlasiakCornell University

MulticutGiven:• 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

s1 s2 s3

t1 t2 t3

Directed Multicut The State of the Art

• Õ(n11/23) - approx algorithm [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.

s1 s2 s3

t1 t2 t3

All approximation algorithms for multicut are based on the LP and compare to the maximum multicommodity flow.

Limited by large integrality gap: Ω(min((k, nδ)) [Saks, Samorodnitsky,Zosin ’04, Chuzhoy, Khanna ’09 ]

Better Lower Bound?Network Coding

• Good News:Coding Rate ≥ Flow Rate, can be a factor k larger

• Bad News:Multicut ≱ Coding Rate, can be a factor k smaller

Results

• Identify a property P of a network code such that any code satisfying P is a lower bound on the multicut.

• Show P is 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

• A network is:• Undirected graph G = (V,E )• Capacity one for each node • Subsets of V: {Si}i ∈[k] , {Ti} i ∈[k] si - ti pairs, i ∈[k], connect to G: • si connects to v ∈Si • v ∈Ti connect to ti

Type of networks giving: Ω(min((k, nδ)) [Saks, Samorodnitsky,Zosin ’04, Chuzhoy, Khanna ’09 ]

Saks et al. Construction

• Begin with The n-path network Pn :v1 v2 v3 v4 v5 vn-1 vn

s1 t1Hypergrid(n, k ) is the k-fold strong product of Pn . It has nk nodes and k s-t pairs and flow rate nk-1Saks et al. show it has multicut at least has knk-1 THEOREMHypergrid(n,k) has a code with rate nk - (n-1)k that is a lower bound on the multicut.

S1 T1

Hypergrid(3,2) = P3 ☒P3

s2

t2

s1

(v2 , v1’ )(v1 , v2’ ) (v3 , v2’ )(v2 , v2’ )

(v2 , v3’ )t1

(v1 , v1’ ) (v3 , v1’ )

(v1 , v3’ ) (v3 , v3’ )

Hypergrid(3,3) = P3 ☒P3 ☒P3

Coding Matrix s2

t2

s1

(v2 , v1’ )(v1 , v2’ ) (v3 , v2’

)(v2 , v2’ )(v2 , v3’ )

t1(v1 , v1’ ) (v3 , v1’

)

(v1 , v3’ ) (v3 , v3’ )

(v1 ,v1’ )(v 2 ,v1’ )

(v3 ,v1’ )

(v1 ,v2’ )(v2 ,v2’ )

(v3 ,v2’ )(v1 ,v3’ )

(v2 ,v3’ )(v3 ,v3’ )

a1 1 1 1 0 0 0 0 0 0b1 0 0 0 1 1 1 0 0 0c1 0 0 0 0 0 0 1 1 1a2 1 0 0 1 0 0 1 0 0b2 0 1 0 0 1 0 0 1 0• Column v describes the linear combination of messages sent by v.

• Column v is a linear combination of columns of predecessors of v.

• ti can decode messages from si.

Columns labeled

with v in V

• Rows labeled with messages• ai , bi , ci originate at si • rate of code = # of messages

a1+a2 b1

a1+b2

a1

c1

b1+b2c1+a

2c1+b2

a2, b2, c2

a1, b1, c1b1+a2

Entries in finite field

Coding Matrix as a Lower Bound

L gives a lower bound on the multicut:For M a minimum multicut, |M| = rank (IM) ≥ rank (LIM) ≥ p.

DEFINITIONA coding matrix L is p-certifiable if 1. For any multicut M, rank(LIM) ≥ p. 2. Column v of L is a linear combination of columns of

incoming sources and predecessors of v that form a clique.

|V|x|M| matrix, column v ∈ M is an indicator vector for v.

Main Theorem*

Given networks N1and N2 with coding matrices L1and L2, there is a coding matrix for N1 ☒N2 :

L = such that:

1. If L1and L2 have rates p1and p2 then L has rate p := n1p2 + n2 p1 - p1p2.2. If L1and L2 are p1and p2 certifiable then L is p-

certifiable.

In1 ⊗ L2L1 ⊗ In2

Saks et al. Construction

• Begin with The n-path network Pn :v1 v2 v3 v4 v5 vn-1 vn

s1 t1Hypergrid(n, k ) is the k-fold strong product of Pn . It has nk nodes and k s-t pairs and flow rate nk-1Saks et al. show it has multicut at least has knk-1 THEOREMHypergrid(n,k) has a code with rate nk -(n-1)k that is a lower bound on the multicut.

Pn is 1-certifiable.

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?