Stable matchings and linear programming

THE stable matching problem was introduced by Gale and Shapley as a model of how to assign students to colleges. It is a classic that has inspired a flood of papers exploring generalizations, variations and applications. For a comprehensive account of these, see Roth and Sotomayor as well as Roth. A stable matching is a matching in a bipartite graph that satisfies additional conditions. Just as we have a linear inequality description of the convex hull of all matchings in a bipartite graph, it is natural to ask if such a description is possible for the convex hull of stable matchings. Vande Vate provided one. His proof (as do subsequent ones) assumes the existence of a stable matching. Existence was established by Gale and Shapley via their proposal algorithm. What has nagged me is that one should be able to obtain existence of a stable matching directly from Vande Vate’s characterization of the convex hull of stable matchings. This article does just that. I associate a linear program with the linear inequality description of the convex hull of stable matchings. I then show that an appropriate dual ascent algorithm for this linear program produces a stable matching. Indeed, the dual ascent algorithm resembles the Gale–Shapley proposal algorithm; which I find satisfying. I begin by introducing notation and stating the stable matching problem. Subsequently, I review the proposal algorithm and Vand Vate’s characterization. This will make the article self-contained.