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Parameterized Complexity of Weighted Multicut in Trees

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Graph-Theoretic Concepts in Computer Science (WG 2022)

Abstract

The Edge Multicut problem is a classical cut problem where given an undirected graph G, a set of pairs of vertices \(\mathcal {P}\), and a budget \(k\), the goal is to determine if there is a set S of at most \(k\) edges such that for each \((s,t) \in \mathcal {P}\), \(G-S\) has no path from s to t. Edge Multicut has been relatively recently shown to be fixed-parameter tractable (FPT), parameterized by \(k\), by Marx and Razgon [SICOMP 2014], and independently by Bousquet et al. [SICOMP 2018]. In the weighted version of the problem, called Weighted Edge Multicut one is additionally given a weight function \(\texttt {wt}: E(G) \rightarrow \mathbb {N}\) and a weight bound \(\textbf{w}\), and the goal is to determine if there is a solution of size at most \(k\) and weight at most \(\textbf{w}\). Both the FPT algorithms for Edge Multicut by Marx et al. and Bousquet et al. fail to generalize to the weighted setting. In fact, the weighted problem is non-trivial even on trees and determining whether Weighted Edge Multicut on trees is FPT was explicitly posed as an open problem by Bousquet et al. [STACS 2009]. In this article, we answer this question positively by designing an algorithm which uses a very recent result by Kim et al. [STOC 2022] about directed flow augmentation as subroutine.

We also study a variant of this problem where there is no bound on the size of the solution, but the parameter is a structural property of the input, for example, the number of leaves of the tree. We strengthen our results by stating them for the more general vertex deletion version.

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Notes

  1. 1.

    Though the formal description of the problem in [15] asks for a solution S with \(\texttt {wt}(S) \le \textbf{w}\), the authors remark that the algorithm in fact finds a minimum weight solution.

  2. 2.

    When dealing with undirected graphs, the flow augmentation restricted to undirected graphs given by Kim et al. [14] may suffice to solve wDPC on undirected graphs. As this problem is not mentioned explicitly in [14], we stick to the directed setting.

  3. 3.

    The dependency in \(k\) is not explicit in [15] but can be easily deduced.

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Acknowledgements

Research supported by the European Research Council (ERC) consolidator grant No. 725978 SYSTEMATICGRAPH. Philipp Schepper is part of Saarbrücken Graduate School of Computer Science, Germany.

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Galby, E., Marx, D., Schepper, P., Sharma, R., Tale, P. (2022). Parameterized Complexity of Weighted Multicut in Trees. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_19

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