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. 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...

• Type: Conference
• Year: 1999
• Where: STOC
• Authors: David R. Karger, Philip N. Klein, Clifford Stein, Mikkel Thorup, Neal E. Young
• Keywords: Algorithms, Geometric Relaxation, Integrality Gap, Multiway Cut, STOC 1999