# Envy-Free Matchings with Lower Quotas

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## Abstract

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 and not all doctor–hospital pairs are acceptable. 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 (in: Procedings of 21st annual ACM-SIAM symposium on discrete algorithms (SODA2010), SIAM, Philadelphia, pp 1235–1253, 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.

## Keywords

Stable matchings Envy-free matchings Lower quotas Polynomial time algorithm Paramodular functions## Notes

### Acknowledgements

I wish to thank the anonymous reviewers whose comments have benefited the paper greatly. I gratefully acknowledge Yasushi Kawase for his helpful comments. This work was supported by JST CREST, Grant Number JPMJCR14D2, Japan.

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