An approximation algorithm for the Generalized k-Multicut problem

Given a graph G = (V, E) with nonnegative costs defined on edges, a positive integer k, and a collection of q terminal sets D = {S1, S2, . . . , Sq}, where each Si is a subset of V (G), the Generalized k-Multicut problem asks to find a set of edges C ⊆ E(G) at the minimum cost such that its removal from G cuts at least k terminal sets in D. A terminal subset Si is cut by C if all terminals in Si are disconnected from one another by removing C from G. This problem is a generalization of the k-Multicut problem and the Multiway Cut problem. The famous Densest k-Subgraph problem can be reduced to the Generalized k-Multicut problem in trees via an approximation preserving reduction. In this paper, we first give an O( √ q)-approximation algorithm for the Generalized k-Multicut problem when the input graph is a tree. The algorithm is based on a mixed strategy of LP-rounding and greedy approach. Moreover, the algorithm is applicable to deal with a class of NP-hard partial optimization problems. As its extensions, we then show that the algorithm can be used to give O( √ q log n)-approximation for the Generalized k-Multicut problem in undirected graphs and O( √ q)-approximation for the k-Forest problem.