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(tlogk) 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(log2k) approximation algorithm for this directed multicut problem. Ifk⪡n, this represents an improvement over theO(lognloglogn) approximation algorithm that is implied by the technique of Seymour.