A New Efficient Algorithm for Weighted Vertex Cover in Bipartite Graphs Based on a Dual Problem

The (un-weighted) vertex cover problem in general graphs is a classical NP-hard problem, but it is polynomial time solvable in bipartite graphs. This paper considers two combinatorial optimization problems. One is the weighted vertex cover problem and the other is the so-called maximum edge packing problem. We proved that in bipartite graphs, maximum edge packing problem can be viewed as the dual of the weighted vertex cover problem, and hence these two problems are polynomial time solvable. We explored the relationships between these two problems in bipartite graphs and some structural results are obtained accordingly. Furthermore, a new efficient algorithm for the weighted vertex cover problem in bipartite graphs is also derived. Our method generalized some previous algorithms for un-weighted vertex cover in bipartite graphs.