An approximation algorithm for the Generalized k-Multicut problem

  • Authors:
  • Peng Zhang;Daming Zhu;Junfeng Luan

  • Affiliations:
  • -;-;-

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2012

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Abstract

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@?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(qlogn)-approximation for the Generalized k-Multicut problem in undirected graphs and O(q)-approximation for the k-Forest problem.