On the Equivalence of the Graph-Structural and Optimization-Based Characterizations of Popular Matchings

Popular matchings provide a model of matching under preferences in which a solution corresponds to a Condorcet winner in voting systems. In a bipartite graph in which the vertices have preferences over their neighbours, a matching is defined to be popular if it does not lose in a majority vote against any matching. In this paper, we study the following three primary problems: only the vertices on one side have preferences; a generalization of this problem allowing ties in the preferences; and the vertices on both sides have preferences. A principal issue in the algorithmic aspects of popular matchings is how to determine the popularity of a matching, because it requires exponential time if the definition is simply applied. In the literature, we have the following two types of characterizations: a graph-structural characterization; and an optimization-based characterization described by maximum-weight matchings. The graph-structural characterizations are specifically designed for each problem and provide a combinatorial structure of the popular matchings. The optimization-based characterizations work in the same manner for all problems, while they do not reveal the structure of the popular matchings. A main contribution of this paper is to provide a direct connection of the above two types of characterizations for all of the three problems. Specifically, we prove that each characterization can be derived from the other, without relying on the fact that they characterize popular matchings. Our proofs offer a comprehensive understanding of the equivalence of the two types of characterizations, and suggest a new interpretation of the graph-structural characterization in terms of the dual optimal solution for the maximum-weight matching problem.