Abstract
In this paper, we consider the k-prize-collecting multicut on a tree (k-PCM(T)) problem. In this problem, we are given an undirected tree T=(V,E), a set of m distinct pairs of vertices P={(s1,t1),…,(sm,tm)} and a parameter k with k≤m. Every edge in E has a nonnegative cost ce. Every pair (si,ti) in P has a nonnegative penalty cost πi. Our goal is to find a multicut M that separates at least k pairs in P such that the total cost, including the edge cost of the multicut M and the penalty cost of the pairs not separated by M, is minimized. This problem generalizes the well-known multicut on a tree problem. Our main work is to present a (4+ε)-approximation algorithm for the k-PCM(T) problem via the methods of primal-dual and Lagrangean relaxation, where ε is any fixed positive number.
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