2026-07-23T19:13:10Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15851 2024-03-06T10:56:21Z ddc:004 open_access

Fairly Popular Matchings and Optimality Kavitha, Telikepalli Bipartite graphs Stable matchings Mixed matchings Polytopes We consider a matching problem in a bipartite graph G = (A ∪ B, E) where vertices have strict preferences over their neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M is at least the number that prefer N; thus M does not lose a head-to-head election against any matching where vertices are voters. It is easy to find popular matchings; however when there are edge costs, it is NP-hard to find (or even approximate) a min-cost popular matching. This hardness motivates relaxations of popularity. Here we introduce fairly popular matchings. A fairly popular matching may lose elections but there is no good matching (wrt popularity) that defeats a fairly popular matching. In particular, any matching that defeats a fairly popular matching does not occur in the support of any popular mixed matching. We show that a min-cost fairly popular matching can be computed in polynomial time and the fairly popular matching polytope has a compact extended formulation. We also show the following hardness result: given a matching M, it is NP-complete to decide if there exists a popular matching that defeats M. Interestingly, there exists a set K of at most m popular matchings in G (where |E| = m) such that if a matching is defeated by some popular matching in G then it has to be defeated by one of the matchings in K. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Telikepalli Kavitha 2022 Is Part Of LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2022.41 urn:nbn:de:0030-drops-158516 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.41 eng https://creativecommons.org/licenses/by/4.0/legalcode