Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut

TL;DR

This paper improves approximation algorithms for the multiway-cut problem using advanced randomized rounding schemes based on geometric relaxations, achieving better performance guarantees than previous methods.

Contribution

It introduces new randomized rounding schemes for geometric relaxations, leading to improved approximation ratios for the multiway-cut problem.

Findings

Achieved a 12/11-approximation for k=3

Developed a 1.3438-approximation for general k

Showed randomized rounding schemes match integrality gaps

Abstract

The multiway-cut problem is, given a weighted graph and k >= 2 terminal nodes, to find a minimum-weight set of edges whose removal separates all the terminals. The problem is NP-hard, and even NP-hard to approximate within 1+delta for some small delta > 0. Calinescu, Karloff, and Rabani (1998) gave an algorithm with performance guarantee 3/2-1/k, based on a geometric relaxation of the problem. In this paper, we give improved randomized rounding schemes for their relaxation, yielding a 12/11-approximation algorithm for k=3 and a 1.3438-approximation algorithm in general. Our approach hinges on the observation that the problem of designing a randomized rounding scheme for a geometric relaxation is itself a linear programming problem. The paper explores computational solutions to this problem, and gives a proof that for a general class of geometric relaxations, there are always randomized rounding schemes that match the integrality gap.