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.