  In the classical partial vertex cover problem, we are given a graph $G$ and
two positive integers $R$ and $L$. The goal is to check whether there is a
subset $V'$ of $V$ of size at most $R$, such that $V'$ covers at least $L$
edges of $G$. The problem is NP-hard as it includes the Vertex Cover problem.
Previous research has addressed the extension of this problem where one has
weight-functions defined on sets of vertices and edges of $G$. In this paper,
we consider the following version of the problem where on the input we are
given an edge-weighted bipartite graph $G$, and three positive integers $R$,
$S$ and $T$. The goal is to check whether $G$ has a subset $V'$ of vertices of
$G$ of size at most $R$, such that the edges of $G$ covered by $V'$ have weight
at least $S$ and they include a matching of weight at least $T$. In the paper,
we address this problem from the perspective of fixed-parameter tractability.
One of our hardness results is obtained via a reduction from the bi-objective
knapsack problem, which we show to be W[1]-hard with respect to one of
parameters. We believe that this problem might be useful in obtaining similar
results in other situations.
