Parameterized Algorithms for Partial Vertex Covers in Bipartite Graphs

In this paper, we discuss parameterized algorithms for variants of the partial vertex cover problem. Recall that in the classical vertex cover problem (VC), we are given a graph [Formula: see text] and a number K and asked if we can cover all of the edges in E, using at most K vertices from V. In the partial vertex cover problem (PVC), in addition to the parameter K, we are given a second parameter [Formula: see text] and the question is whether we can cover at least [Formula: see text] of the edges in E using at most K vertices from V. The weighted generalizations of the VC and PVC problems are called the weighted vertex cover (WVC) and the partial weighted vertex cover problem (WPVC) respectively. In the WPVC problem, we are given two parameters R and L, associated respectively with the vertex set V and edge set E of the graph [Formula: see text]. 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. (In the WVC problem, the goal is to cover all the edges with vertices whose total weight is at most R). This paper studies several variants of the PVC problem 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 constraint and show that it is fixed-parameter tractable for a certain class of graphs.