, Volume 27, Issue 1, pp 65-77

On the generalized multiway cut in trees problem

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Abstract

Given a tree $T = (V, E)$ with $n$ vertices and a collection of terminal sets $D = \{S_1, S_2, \ldots , S_c\}$ , where each $S_i$ is a subset of $V$ and $c$ is a constant, the generalized multiway cut in trees problem (GMWC(T)) asks to find a minimum size edge subset $E^{\prime } \subseteq E$ such that its removal from the tree separates all terminals in $S_i$ from each other for each terminal set $S_i$ . The GMWC(T) problem is a natural generalization of the classical multiway cut in trees problem, and has an implicit relation to the Densest $k$ -Subgraph problem. In this paper, we show that the GMWC(T) problem is fixed-parameter tractable by giving an $O(n^2 + 2^k)$ time algorithm, where $k$ is the size of an optimal solution, and the GMWC(T) problem is polynomial time solvable when the problem is restricted in paths.We also discuss some heuristics for the GMWC(T) problem

A preliminary version of this paper appeared in the Proceedings of the 6th International Conference of Combinatorial Optimization and Applications (COCOA 2012) (Liu and Zhang 2012).