Perfect Matchings and Popularity in the Many-to-Many Setting
Abstract
We consider a matching problem in a bipartite graph 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 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 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.
Keywords
Cite
@article{arxiv.2411.00384,
title = {Perfect Matchings and Popularity in the Many-to-Many Setting},
author = {Telikepalli Kavitha and Kazuhisa Makino},
journal= {arXiv preprint arXiv:2411.00384},
year = {2024}
}