Multicut Lower Bounds via Network Coding

TL;DR

This paper presents a novel method using network coding to establish lower bounds on multicut size, identifying a class of networks where coding rate equals the minimum multicut, and applying this to strengthen existing bounds.

Contribution

It introduces a new technique linking network coding rates to multicut bounds and characterizes a class of networks where these quantities are equal, enhancing multicut lower bound analysis.

Findings

Identified a class of networks where network coding rate equals the multicut size.

Strengthened multicut lower bounds using the identified network class.

Provided an optimal network coding solution matching the multicut.

Abstract

We introduce a new technique to certify lower bounds on the multicut size using network coding. In directed networks the network coding rate is not a lower bound on the multicut, but we identify a class of networks on which the rate is equal to the size of the minimum multicut and show this class is closed under the strong graph product. We then show that the famous construction of Saks et al. that gives a $\Theta(k)$ gap between the multicut and the multicommodity flow rate is contained in this class. This allows us to apply our result to strengthen their multicut lower bound, determine the exact value of the minimum multicut, and give an optimal network coding solution with rate matching the multicut.