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Stability Notions for Hospital Residents with Sizes

Authors Haricharan Balasundaram , J. B. Krishnashree , Girija Limaye , Meghana Nasre

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Stable matchings
  • Hospital Residents with sizes
  • Approximation algorithms
  • NP-hardness

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Abstract

The Hospital Residents problem with sizes (HRS) is a generalisation of the well-studied hospital residents (HR) problem. In the HRS problem, an agent a has a size s(a) and the agent occupies s(a) many positions of the hospital h when assigned to h. The notion of stability in this setting is suitably modified, and it is known that deciding whether an HRS instance admits a stable matching is NP-hard under severe restrictions. In this work, we explore a variation of stability, which we term occupancy-based stability. This notion was defined by McDermid and Manlove (J. of Comb. Opt. 2010) but remained unexplored to the best of our knowledge. In our work, we show that every HRS instance admits an occupancy-stable matching. We further show that computing a maximum-size occupancy-stable matching is NP-hard. We complement our hardness result by providing an approximation algorithm with a guarantee strictly better than 3 for the max-size occupancy-stable matching problem.
Given that the classical notion of stability adapted for HRS is not guaranteed to exist in general, we show a practical restriction under which a stable matching is guaranteed to exist. We present an efficient algorithm to output a stable matching in the restricted HRS instances. We also provide an alternate NP-hardness proof for the decision version of the stable matching problem for HRS which imposes a severe restriction on the number of neighbours of non-unit sized agents.

Cite As Get BibTex

Haricharan Balasundaram, J. B. Krishnashree, Girija Limaye, and Meghana Nasre. Stability Notions for Hospital Residents with Sizes. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.11

Author Details

Haricharan Balasundaram
  • Indian Institute of Technology Madras, Chennai, India
J. B. Krishnashree
  • Tata Institute of Fundamental Research Mumbai, India
Girija Limaye
  • Birla Institute of Technology and Science, Pilani, K K Birla Goa Campus, India
Meghana Nasre
  • Indian Institute of Technology Madras, Chennai, India

Acknowledgements

We thank the anonymous reviewers for their comments that significantly improved the presentation of the paper.

References

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