Approximation Algorithms for Steiner and Directed Multicuts

In this paper we consider theSteiner multicutproblem. This is a generalization of the minimum multicut problem where instead of separating nodepairs, the goal is to find a minimum weight set of edges that separates all givensetsof nodes. A set is considered separated if it is not contained in a single connected component. We show anO(log³(kt)) approximation algorithm for the Steiner multicut problem, wherekis the number of sets andtis the maximum cardinality of a set. This improves theO(t log k) bound that easily follows from the previously known multicut results. We also consider an extension of multicuts to directed case, namely the problem of finding a minimum-weight set of edges whose removal ensures that none of the strongly connected components includes one of the prespecifiedknode pairs. In this paper we describe anO(log² k) approximation algorithm for this directed multicut problem. Ifk ? n, this represents an improvement over theO(log n log log n) approximation algorithm that is implied by the technique of Seymour.