Approximating Steiner k-Cuts

Abstract

We consider the Steiner k-cut problem, which is a common generalization of the k-cut problem and the multiway cut problem: given an edge-weighted undirected graph G = (V,E), a subset of vertices X $\subseteq$ V called terminals, and an integer k ≤ |X|, the objective is to find a minimum weight set of edges whose removal results in k disconnected components, each of which contains at least one terminal. We give two approximation algorithms for the problem: a 2 − 2/k-approximation based on Gomory-Hu trees, and a 2 − 2/|X|-approximation based on LP rounding. The latter algorithm is based on rounding a generalization of a linear programming relaxation suggested by Naor and Rabani [8]. The rounding uses the Goemans and Williamson primal-dual algorithm (and analysis) for the Steiner tree problem [4] in an interesting way and differs from the rounding in [8]. We use the insight from the rounding to develop an exact bi-directed formulation for the global minimum cut problem (the k-cut problem with k = 2).