Envy-free Matchings with Lower Quotas

Author Yu Yokoi

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Yu Yokoi

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Yu Yokoi. Envy-free Matchings with Lower Quotas. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 67:1-67:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


While every instance of the Hospitals/Residents problem admits a stable matching, the problem with lower quotas (HR-LQ) has instances with no stable matching. For such an instance, we expect the existence of an envy-free matching, which is a relaxation of a stable matching preserving a kind of fairness property. In this paper, we investigate the existence of an envy-free matching in several settings, in which hospitals have lower quotas. We first provide an algorithm that decides whether a given HR-LQ instance has an envy-free matching or not. Then, we consider envy-freeness in the Classified Stable Matching model due to Huang (2010), i.e., each hospital has lower and upper quotas on subsets of doctors. We show that, for this model, deciding the existence of an envy-free matching is NP-hard in general, but solvable in polynomial time if quotas are paramodular.
  • stable matchings
  • envy-free matchings
  • lower quotas
  • polynomial time algorithm
  • paramodular functions


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