Parameterized Complexity of Weighted Multicut in Trees

Abstract

The Edge Multicut problem is a classical cut problem where given anundirected 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 suchthat for each $(s,t) \in \mathcal{P}$, $G-S$ has no path from $s$ to $t$. EdgeMulticut has been relatively recently shown to be fixed-parameter tractable(FPT), parameterized by $k$, by Marx and Razgon [SICOMP 2014], andindependently by Bousquet et al. [SICOMP 2018]. In the weighted version of theproblem, called Weighted Edge Multicut one is additionally given a weightfunction $\mathtt{wt} : E(G) \to \mathbb{N}$ and a weight bound $w$, and thegoal is to determine if there is a solution of size at most $k$ and weight atmost $w$. Both the FPT algorithms for Edge Multicut by Marx et al. and Bousquetet al. fail to generalize to the weighted setting. In fact, the weightedproblem is non-trivial even on trees and determining whether Weighted EdgeMulticut on trees is FPT was explicitly posed as an open problem by Bousquet etal. [STACS 2009]. In this article, we answer this question positively bydesigning an algorithm which uses a very recent result by Kim et al. [STOC2022] about directed flow augmentation as subroutine. We also study a variant of this problem where there is no bound on the sizeof the solution, but the parameter is a structural property of the input, forexample, the number of leaves of the tree. We strengthen our results by statingthem for the more general vertex deletion version.<br