Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties

Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching due to the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced in preference lists, this is not the case because the number of residents may vary over stable matchings. In this paper, we formulate an optimization problem that asks to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios. We provide exact values of these maximum gaps in all scenarios. Subsequently, we propose a strategy-proof approximation algorithm for our problem; in one scenario it solves the problem optimally, and in the other three scenarios, which are NP-hard, it yields a better approximation factor than a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios.