  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 $\mathtt{wt} : E(G) \to \mathbb{N}$ 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 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.

Data Structures and Algorithms;Computational Complexity