The multi-multiway cut problem

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 S1,S2,…,Sk of vertices. The goal is to find a subset of edges of minimum total weight whose removal completely disconnects each one of the sets S1,S2,…,Sk, i.e., disconnects every pair of vertices u and v such that u,v∈Si, for some i. This problem generalizes both the multicut problem, when |Si|=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(logk) 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(logk) when restricted to such “light” instances. Specifically, we obtain an -approximation algorithm for the problem when all edge weights are at least 1 (here denotes the value of a natural linear programming relaxation of the problem). The latter improves the approximation ratio for the minimum multicut problem (implied by the work of Seymour and Even et al.).