An approximation algorithm for the Generalized -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 = { S 1 , S 2 , ... , S q } , where each S i is a subset of V ( G ) , the Generalized k -Multicut problem asks to find a set of edges C [subE] E ( G ) at the minimum cost such that its removal from G cuts at least k terminal sets in D . A terminal subset S i is cut by C if all terminals in S i 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.