In this chapter we consider two very closely related problems, maximum weighted matching and minimum cost vertex cover in bipartite graphs. Linear programming duality plays a crucial role in understanding the relationship between these problems. We will show that the natural linear programming relaxations for both the matching problem and the vertex cover problem are integral, and then use duality to obtain a min–max relation between them. Nevertheless, our proofs of integrality use the iterative method by arguing the existence of 1-elements in an extreme point solution.

In the first section, we show the integrality of the more standard maximization version of the matching problem. In the following sections, we show two applications of the proof technique for integrality to derive approximation results for NP-hard problems. We first present a new proof of an approximation result for the generalized assignment problem and then present an approximation result for the budgeted allocation problem. The proofs of both of these results develop on the integrality result for the bipartite matching problem and introduce the iterative relaxation method. Following this, we discuss the integrality of the bipartite vertex cover problem formulation and conclude with a short section on the duality relation between these problems and some historical notes.

Matchings in bipartite graphs

In this section, we show that the matching polytope in bipartite graphs is integral. Given a bipartite graph G = (V1 ∪ V2, E) and a weight function w: E → ℛ, the maximum matching problem is to find a set of vertex-disjoint edges of maximum total weight.