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, ..., 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 S₁,S₂, ..., S_k, i.e., disconnects every pair of vertices u and u 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 [GVY96] on the standard multicut problem. Namely, our algorithm has an O(log 2k) 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 2k) 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, where LP is the value of a natural LP-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 [Sey95] and Even et al. [ENSS98]).