  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.
