(2022) Parameterized Complexity of Weighted Multicut in Trees.
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Abstract
The \textsc{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 $t$. \textsc{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 \textsc{Weighted Edge Multicut} one is additionally given a weight function $\texttt{wt} : E(G) \to \mathbb{N}$ and a weight bound $w$, and the goal is to determine if there is a solution of size at most $k$ and weight at most $w$. Both the FPT algorithms for \textsc{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 \textsc{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.
| Item Type: | Conference or Workshop Item (A Paper) (Paper) |
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| Divisions: | Dániel Marx (DM) |
| Conference: | WG International Workshop on Graph-Theoretic Concepts in Computer Science |
| Depositing User: | Philipp Schepper |
| Date Deposited: | 13 Oct 2022 09:57 |
| Last Modified: | 17 Oct 2022 05:50 |
| Primary Research Area: | NRA1: Trustworthy Information Processing |
| URI: | https://publications.cispa.saarland/id/eprint/3840 |
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