On the fixed-parameter tractability of the partial vertex cover problem with a matching constraint in edge-weighted bipartite graphs

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.