Parameterized Complexity of Weighted Multicut in Trees

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.