# 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.

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