Given a weighted undirected graph, the maximum matching problem is to find a matching with maximum total weight. In his seminal paper, Edmonds [35] described an integral polytope for the matching problem, and the famous Blossom Algorithm for solving the problem in polynomial time.

In this chapter, we will show the integrality of the formulation given by Edmonds [35] using the iterative method. The argument will involve applying uncrossing in an involved manner and hence we provide a detailed proof. Then, using the local ratio method, we will show how to extend the iterative method to obtain approximation algorithms for the hypergraph matching problem, a generalization of the matching problem to hypergraphs.

Graph matching

Matchings in bipartite graphs are considerably simpler than matchings in general graphs; indeed, the linear programming relaxation considered in Chapter 3 for the bipartite matching problem is not integral when applied to general graphs. See Figure 9.1 for a simple example.

Linear programming relaxation

Given an undirected graph G = (V, E) with a weight function w: E → ℛ on the edges, the linear programming relaxation for the maximum matching problem due to Edmonds is given by the following LPM(G). Recall that E(S) denotes the set of edges with both endpoints in S ⊆ V and x(F) is a shorthand for ∑e∈Fxe for F ⊆ E.

Although there are exponentially many inequalities in LPM(G), there is an efficient separation oracle for this linear program, obtained by Padberg and Rao using Gomory-Hu trees.