{"type":"ScholarlyArticle","title":"Couples Can Be Tractable: New Algorithms and Hardness Results for the Hospitals/Residents Problem with Couples","authors":[{"name":"Gergely Csáji","slug":"gergely-csaji","given":"Gergely","family":"Csáji","affiliation":"ELTE Eötvös Loránd University , Budapest , Hungary; HUN-REN KRTK KTI , Budapest , Hungary"},{"name":"David Manlove","slug":"david-manlove","given":"David","family":"Manlove","affiliation":"School of Computing Science , University of Glasgow , Glasgow G12 8QQ , UK"},{"name":"Iain McBride","slug":"iain-mcbride","given":"Iain","family":"McBride","affiliation":"School of Computing Science , University of Glasgow , Glasgow G12 8QQ , UK"},{"name":"James Trimble","slug":"james-trimble","given":"James","family":"Trimble","affiliation":"School of Computing Science , University of Glasgow , Glasgow G12 8QQ , UK"}],"abstract":"In this paper we study the Hospitals/Residents problem with Couples (HRC), where a solution is a stable matching or a report that none exists. \r\nWe present a novel polynomial-time algorithm that can find a near-feasible stable matching (adjusting the hospitals' capacities by at most 1) in an HRC instance where the couples' preferences are sub-responsive (i.e., if one member switches to a better hospital, than the couple also improves if the new pair is also acceptable) and sub-complete (i.e., each pair of hospitals that are individually acceptable to both members are jointly acceptable for the couple) by reducing it to an instance of the Stable Fixtures problem.  We also present a polynomial-time algorithm for HRC in a sub-responsive, sub-complete instance that is a Dual Market, or where all couples are one of several possible types. \r\nOur polynomial-time solvability results greatly expand the class of known tractable instances of HRC.\r\nWe complement our algorithms with several hardness results. We show that HRC with sub-responsive and sub-complete couples is NP-hard, even with other strong restrictions. We also show that HRC with a Dual Market is NP-hard under several simultaneous restrictions.","doi":"10.24963/ijcai.2024/302","url":"https://ijcai.org/proceedings/2024/302","pdf":"https://ijcai.org/proceedings/2024/0302.pdf","markdown":"https://ijcai.org/proceedings/2024/302.md","bibtex":"https://ijcai.org/proceedings/2024/bibtex/302","volume":{"slug":"2024","title":"Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2024","url":"https://ijcai.org/proceedings/2024/","year":2024,"location":"Jeju, Korea","dates":"3-9 August 2024","editors":"Kate Larson","isbn":"978-1-956792-04-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2024"},"section":"Game Theory and Economic Paradigms","track":"Main Track","number":302,"pages":{"first":2731,"last":2739},"keywords":["Game Theory and Economic Paradigms: GTEP: Computational social choice","Game Theory and Economic Paradigms: GTEP: Mechanism design"],"videos":[],"references":[{"position":1,"raw":"[Biró and Klijn, 2013] P. 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