(2022) Parameterized Complexity of Weighted Multicut in Trees.

Text (Full Version)
2205.10105.pdf  Published Version Download (869kB)  Preview 
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 $(s,t) \in \mathcal{P}$, $GS$ has no path from $s$ to $t$. \textsc{Edge Multicut} has been relatively recently shown to be fixedparameter 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 nontrivial 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) 

Divisions:  Dániel Marx (DM) 
Conference:  WG International Workshop on GraphTheoretic 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 
Actions
Actions (login required)
View Item 