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 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.

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