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 $S_1,S_2,\dots ,S_k$ of vertices. The goal is to find a subset of edges of minimum total weight whose removal completely disconnects each one of the sets $S_1,S_2,\dots ,S_k$, i.e., disconnects every pair of vertices $u$ and $v$ such that $u,v\in S_i$, for some $i$. This problem generalizes both the multicut problem, when $\left|S_i\right|=2$, for $1\le i\le k$, 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 $O(log\mathrm{LP})$-approximation algorithm for the problem when all edge weights are at least 1 (here $\mathrm{LP}$ denotes the value of a natural linear programming relaxation of the problem). The latter improves the $O(log\mathrm{LP}loglog\mathrm{LP})$ approximation ratio for the minimum multicut problem (implied by the work of Seymour and Even et al.).