Gergely Csáji, Tamás Király, Kenjiro Takazawa, Yu Yokoi
{"title":"Popular Maximum-Utility Matchings with Matroid Constraints","authors":"Gergely Csáji, Tamás Király, Kenjiro Takazawa, Yu Yokoi","doi":"arxiv-2407.09798","DOIUrl":null,"url":null,"abstract":"We investigate weighted settings of popular matching problems with matroid\nconstraints. The concept of popularity was originally defined for matchings in\nbipartite graphs, where vertices have preferences over the incident edges.\nThere are two standard models depending on whether vertices on one or both\nsides have preferences. A matching $M$ is popular if it does not lose a\nhead-to-head election against any other matching. In our generalized models,\none or both sides have matroid constraints, and a weight function is defined on\nthe ground set. Our objective is to find a popular optimal matching, i.e., a\nmaximum-weight matching that is popular among all maximum-weight matchings\nsatisfying the matroid constraints. For both one- and two-sided preferences\nmodels, we provide efficient algorithms to find such solutions, combining\nalgorithms for unweighted models with fundamental techniques from combinatorial\noptimization. The algorithm for the one-sided preferences model is further\nextended to a model where the weight function is generalized to an\nM$^\\natural$-concave utility function. Finally, we complement these\ntractability results by providing hardness results for the problems of finding\na popular near-optimal matching. These hardness results hold even without\nmatroid constraints and with very restricted weight functions.","PeriodicalId":501316,"journal":{"name":"arXiv - CS - Computer Science and Game Theory","volume":"68 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Computer Science and Game Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.09798","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate weighted settings of popular matching problems with matroid
constraints. The concept of popularity was originally defined for matchings in
bipartite graphs, where vertices have preferences over the incident edges.
There are two standard models depending on whether vertices on one or both
sides have preferences. A matching $M$ is popular if it does not lose a
head-to-head election against any other matching. In our generalized models,
one or both sides have matroid constraints, and a weight function is defined on
the ground set. Our objective is to find a popular optimal matching, i.e., a
maximum-weight matching that is popular among all maximum-weight matchings
satisfying the matroid constraints. For both one- and two-sided preferences
models, we provide efficient algorithms to find such solutions, combining
algorithms for unweighted models with fundamental techniques from combinatorial
optimization. The algorithm for the one-sided preferences model is further
extended to a model where the weight function is generalized to an
M$^\natural$-concave utility function. Finally, we complement these
tractability results by providing hardness results for the problems of finding
a popular near-optimal matching. These hardness results hold even without
matroid constraints and with very restricted weight functions.