The multi-multiway cut problem

Abstract

In this paper, we define and study a natural generalization of the multicut and multiway cut problems: the minimum multi-multiway cut problem. The input to the problem is a weighted undirected graph G = ( V , E ) and k sets S 1 , S 2 , … , S k of vertices. The goal is to find a subset of edges of minimum total weight whose removal completely disconnects each one of the sets S 1 , S 2 , … , S k , i.e., disconnects every pair of vertices u and v such that u , v ∈ S i , for some i . This problem generalizes both the multicut problem, when | S i | = 2 , for 1 ≤ i ≤ k , and the multiway cut problem, when k = 1 . We present an approximation algorithm for the multi-multiway cut problem with an approximation ratio which matches that obtained by Garg, Vazirani, and Yannakakis on the standard multicut problem. Namely, our algorithm has an O ( log k ) approximation ratio. Moreover, we consider instances of the minimum multi-multiway cut problem which are known to have an optimal solution of light weight. We show that our algorithm has an approximation ratio substantially better than O ( log k ) when restricted to such “light” instances. Specifically, we obtain an O ( log LP ) -approximation algorithm for the problem when all edge weights are at least 1 (here LP denotes the value of a natural linear programming relaxation of the problem). The latter improves the O ( log LP log log LP ) approximation ratio for the minimum multicut problem (implied by the work of Seymour and Even et al.).