Rounding algorithms for a geometric embedding of minimum multiway cut

Abstract

Given an undirected graph with edge costs and a subset of k≥3 nodes called terminals, a multiway, or k-way, cut is a subset of the edges whose removal disconnects each terminal from the others. The multiway cut problem is to find a minimum-cost multiway cut. This problem is Max-SNP hard. Recently Calinescu, Karloff, and Rabani (STOC'98) gave a novel geometric relaxation of the problem and a rounding scheme that produced a (3/2-1/k)-approximation algorithm. In this paper, we study their geometric relaxation. In particular, we study the worst-case ratio between the value of the relaxation and the value of the minimum multicut (the so-called integrality gap of the relaxation). For k = 3, we show the integrality gap is 12/11, giving tight upper and lower bounds. That is, we exhibit a graph with integrality gap 12/11 and give an algorithm that finds a cut of value 12/11 times the relaxation value. This is the best possible performance guarantee for any algorithm based purely on the value of the relaxation and improves on Calinescu et al.'s factor of 7/6. We also improve the upper bounds for all larger values of k. For k = 4, 5, our best upper bounds are based on computer constructed and analyzed rounding schemes, while for k>6 we give an algorithm with performance ratio 1.3438-εk. Our results were discovered with the help of computational experiments that we also describe here.