Optimal Capacity Modification for Stable Matchings with Ties

[2411.10284] Optimal Capacity Modification for Stable Matchings with Ties Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Data Structures and Algorithms arXiv:2411.10284 (cs) [Submitted on 15 Nov 2024 ( v1 ), last revised 14 Apr 2026 (this version, v4)] Title: Optimal Capacity Modification for Stable Matchings with Ties Authors: Keshav Ranjan , Meghana Nasre , Prajakta Nimbhorkar View a PDF of the paper titled Optimal Capacity Modification for Stable Matchings with Ties, by Keshav Ranjan and 2 other authors View PDF HTML (experimental) Abstract: We consider the Hospitals/Residents (HR) problem in the presence of ties in hospital preferences. Among the three notions of stability, namely weak stability, strong stability, and super-stability, we focus on the notion of strong stability. Strong stability has many desirable properties, both theoretically and in practice; however, its existence is not guaranteed. In this paper, our objective is to optimally increase the quotas of hospitals to ensure that a strongly stable matching exists in the modified instance. We explore two natural optimization criteria: (i) minimizing the total capacity increase across all hospitals (MINSUM) and (ii) minimizing the maximum capacity increase for any hospital (MINMAX). We show that the MINSUM problem admits a polynomial-time algorithm. We also establish an analog of the well-known rural hospitals theorem [Gale & Sotomayor, 1985; Roth, 1986], adapted to the MINSUM augmentation setting. We consider a generalization of the MINSUM problem in which each hospital incurs a cost per unit increase in its quota. We show that the cost version of the MINSUM problem is NP-hard and inapproximable within any multiplicative factor, even if the costs are zero or one. For the MINSUM objective with a set of forced edges, we give a polynomial-time algorithm. In contrast to the above results for the MINSUM problem, we show that the MINMAX problem is NP-hard. When hospital preference lists have ties of length at most $\ell+1$, we give a polynomial-time algorithm that increases each hospital's quota by at most $\ell$, ensuring the resulting instance admits a strongly stable matching. Moreover, among all such augmentations, our algorithm outputs the best strongly stable matching from the residents' perspective. Comments: 33 pages, 6 figures Subjects: Data Structures and Algorithms (cs.DS) ; Computer Science and Game Theory (cs.GT) Cite as: arXiv:2411.10284 [cs.DS] (or arXiv:2411.10284v4 [cs.DS] for this version) https://doi.org/10.48550/arXiv.2411.10284 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Keshav Ranjan [ view email ] [v1] Fri, 15 Nov 2024 15:40:45 UTC (49 KB) [v2] Wed, 25 Dec 2024 12:21:43 UTC (52 KB) [v3] Fri, 23 May 2025 10:10:22 UTC (60 KB) [v4] Tue, 14 Apr 2026 06:09:55 UTC (59 KB) Full-text links: Access Paper: View a PDF of the paper titled Optimal Capacity Modification for Stable Matchings with Ties, by Keshav Ranjan and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.DS < prev | next > new | recent | 2024-11 Change to browse by: cs cs.GT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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