Approximation Algorithms for the Bipartite Multi-cut Problem

TL;DR

This paper introduces the Bipartite Multi-cut problem, proves its NP-hardness, and develops LP and SDP approximation algorithms leveraging advanced mathematical techniques.

Contribution

It generalizes several cut problems and provides novel LP and SDP approximation algorithms with theoretical guarantees.

Findings

The Bipartite Multi-cut problem is NP-hard.

LP and SDP algorithms achieve approximation guarantees.

The SDP algorithm uses the Structure Theorem of ℓ₂² metrics.

Abstract

We introduce the {\it Bipartite Multi-cut} problem. This is a generalization of the {\it st-Min-cut} problem, is similar to the {\it Multi-cut} problem (except for more stringent requirements) and also turns out to be an immediate generalization of the {\it Min UnCut} problem. We prove that this problem is {\bf NP}-hard and then present LP and SDP based approximation algorithms. While the LP algorithm is based on the Garg-Vazirani-Yannakakis algorithm for {\it Multi-cut}, the SDP algorithm uses the {\it Structure Theorem} of $\ell_2^2$ Metrics.