A computational study of a geometric embedding of minimum multiway cut

Abstract

In the minimum multiway cut problem, the goal is to find a minimum cost set of edges whose removal disconnects a certain set of k distinguished vertices in a graph. The problem is MAX-SNP hard for k >/= 3. Clinescu, Karloff, and Rabani gave a geometric relaxation of the problem and a rounding scheme, to produce an approximation algorithm that has a performance guarantee of 3/2 - 1/k. In a subsequent study, Karger, Klein, Stein, Thorup, and Young discovered improved rounding schemes via computation experiments for various values of k, yielding approximation algorithms with improved performance guarantees. Their rounding scheme for k = 3 is provably optimal (i.e., its performance guarantee is equal to the integrality gap of the relaxation), but their rounding schemes for k > 3 seemed unlikely to be optimal. In the present work, we improve these rounding schemes for small values of k > 3, yielding improved approximation algorithms. These improvements were discovered by applying an improved analysis to the same set of computational experiments used by Karger et al.

(cont.) We also present computer-aided proofs of improved lower bounds on the integrality gap for various values of k > 3. For the k = 4 case, for instance, our work demonstrates a lower and upper bound of 1.1052 and 1.1494, respectively, improving upon the previously best known bounds of 1.0909 and 1.1539. Finally, we present additional computational experiments that may shed some light on the nature of the optimal rounding scheme for the k = 4 case.