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.
Highlights
Since the seminal work of Gale and Shapley [11], the Hospitals/Residents problem (HR, for short), or the College Admission problem, has been studied extensively [14, 20, 27]
For an Hospitals/Residents problem with lower quotas (HR-LQ) instance, suppose that we find a stable matching while disregarding the lower quotas, and that the obtained matching does not meet the lower quotas
If we find a stable matching that meets the lower quotas after repeating such adjustments, the obtained matching is an envy-free matching of the original instance
Summary
Since the seminal work of Gale and Shapley [11], the Hospitals/Residents problem (HR, for short), or the College Admission problem, has been studied extensively [14, 20, 27]. (See “Related Works” below for results on stable matchings of CSM and its generalizations.) In Huang’s original model, each hospital has a family of sets of doctors, called classes, and each class has an upper and a lower quota We formulate this setting by letting each hospital have a pair of set functions defined on the set of acceptable doctors. We show that if the pair of lower and upper quota functions of each hospital is paramodular [9] (see Section 4 for the definition), we can decide the existence of an envy-free matching in polynomial time This means that the problem is tractable if the family of acceptable doctor sets forms a generalized matroid for each hospital. Because constraints defined on a laminar (or hierarchical) family yield a generalized matroid, our tractable special case includes a case in which each hospital defines quotas on a laminar family of doctors
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