On the Partial Vertex Cover Problem in Bipartite Graphs - a Parameterized Perspective

In this paper, we examine variants of the partial vertex cover problem from the perspective of parameterized algorithms. Recall that in the classical vertex cover problem (VC), we are given a graph \(\mathbf{G = \langle V, E \rangle }\) and a number k and asked if we can cover all of the edges in \(\textbf{E}\), using at most k vertices from \(\textbf{V}\). The partial vertex cover problem (PVC) is a more general version of the VC problem in which we are given an additional parameter \(k'\). We then ask the question of whether at least \(k'\) of the edges in \(\textbf{E}\) can be covered using at most k vertices from \(\textbf{V}\). Note that the VC problem is a special case of the PVC problem when \(k'=|\textbf{E}|\). In this paper, we study the weighted generalizations of the PVC problem. This is called the weighted partial vertex cover problem (WPVC). In the WPVC problem, we are given two parameters R and L, associated respectively with the vertex set \(\textbf{V}\) and edge set \(\textbf{E}\) of the graph \(\textbf{G}\) respectively. Additionally, we are given non-negative integral weight functions for the vertices and the edges. The goal then is to cover edges of total weight at least L, using vertices of total weight at most R. This paper studies several variants of the PVC and WPVC problems and establishes new results from the perspective of fixed-parameter tractability and W[1]-hardness. We also introduce a new problem called the partial vertex cover with matching constraints and show that it is Fixed-Parameter Tractable (FPT) for a certain class of graphs. Finally, we show that the WPVC problem is APX-complete for bipartite graphs.