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 k1 and k2 . The goal is to check whether there is a subset V′ of V of size at most k1, such that V′ covers at least k2 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 as the input we are given an edge-weighted bipartite graph G with weights from N, and three positive integers k1, k2 and k3 . The goal is to check whether G has a subset V′ of vertices of G of size at most k1, such that the edges of G covered by V′ have weight at least k2 and they include a matching of weight at least k3 . In the paper, we address this problem from the perspective of fixed-parameter tractability and algorithms. We present some W[1]-hardness, paraNP-hardness results for our problem. On the positive side, we show that the problem is fixed-parameter tractable with respect to certain parameters. One of our W[1]-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 the parameters. We believe that this problem might be useful in obtaining similar results in other situations. © 2022, Brown University. All rights reserved.