Rounding algorithms for a geometric embedding of minimum multiway cut
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. T his 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 parti cular, 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 family of graphs with integrality gaps arbit rarily close to 12/11 and give an algorithm that finds a cut of value 12/11 times the relaxation value. Our lower bound shows that this is the best