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
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Acknowledgments
We are very grateful to the anonymous referees for their valuable comments which help to improve the presentation of the paper. Hong Liu is supported by the Independent Innovation Foundation of Shandong University (2012TS071). Peng Zhang is supported by the National Natural Science Foundation of China (60970003), China Postdoctoral Science Foundation (200902562), the Special Foundation of Shandong Province Postdoctoral Innovation Project (200901010), and the Independent Innovation Foundation of Shandong University (2012TS072).
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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).
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Liu, H., Zhang, P. On the generalized multiway cut in trees problem. J Comb Optim 27, 65–77 (2014). https://doi.org/10.1007/s10878-012-9565-9
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DOI: https://doi.org/10.1007/s10878-012-9565-9