Perfect Matchings and Popularity in the Many-to-Many Setting

TL;DR

This paper introduces a polynomial-time algorithm to find minimum-cost popular perfect matchings in a bipartite graph with capacities and preferences, generalizing known one-to-one results to many-to-many settings.

Contribution

It extends the characterization of popular perfect matchings to many-to-many bipartite graphs and provides an efficient algorithm for min-cost solutions.

Findings

Existence of popular perfect matchings in the many-to-many setting

Efficient polynomial-time algorithm for min-cost popular perfect matchings

Generalization of stable matching characterization to many-to-many scenarios

Abstract

We consider a matching problem in a bipartite graph $G$ where every vertex has a capacity and a strict preference order on its neighbors. Furthermore, there is a cost function on the edge set. We assume $G$ admits a perfect matching, i.e., one that fully matches all vertices. It is only perfect matchings that are feasible for us and we are interested in those perfect matchings that are popular within the set of perfect matchings. It is known that such matchings (called popular perfect matchings) always exist and can be efficiently computed. What we seek here is not any popular perfect matching, but a min-cost one. We show a polynomial-time algorithm for finding such a matching; this is via a characterization of popular perfect matchings in $G$ in terms of stable matchings in a colorful auxiliary instance. This is a generalization of such a characterization that was known in the one-to-one setting.