Abundant Neighborhoods, Two‐Sided Markets, and Maximal Matchings
提出图论新性质“富邻域”，用于统一分析稳定匹配、最大权重匹配等双边市场变体，并基于此开发最小最大匹配问题的新整数规划模型，实验表明求解稠密图（500顶点）比现有方法快30%-50%。
ABSTRACT I introduce a new graph‐theoretic property called abundant neighborhoods . This property is motivated by studying the thickness of economic markets. A vertex is, roughly, guaranteed to match if and only if it has an abundant neighborhood. This fact holds across numerous variants of two‐sided markets that are studied across the economics, operations research, and computer science literature. I introduce a new formalism to study these variants under a unifying framework, which I call matching rules , allowing us to study hitherto different types of markets (equivalently, graph matching problems) together. In particular, stable matchings, max‐weight matchings, and rank‐maximal matchings can be studied together as they are all surjective maximal matching rules. Lastly, the abundant neighborhood property can be used to study properties of maximal matchings: a vertex is matched in all maximal matchings if and only if it has an abundant neighborhood. With this observation, I develop a novel decomposition for studying maximal matchings. I use it to introduce a new integer programming formulation of the minimum maximal matching problem, which represents the worst‐case performance of a market. I show with experiments that using this formulation solves the problem for dense graphs with up to 500 vertices with 30%–50% less time than previous approaches from the literature and yields much tighter solutions for timed‐out instances.