最大化满足医院/住院医生约束下的最低要求问题
Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties
摘要：
针对农村地区医院住院医师短缺的严重问题，本文研究了医院/住院医师模型，在该模型中，医院与较低的配额相关联，目标是尽可能地满足他们。当偏好列表是严格的时，由于众所周知的农村医院定理，在任何稳定匹配中分配给每个医院的居民数量都是相同的;因此没有算法干预的空间。然而，当偏好列表中引入关系时，这将不再适用，因为居民的数量可能会随着稳定匹配而变化。在本文中，我们制定了一个优化问题，以找到一个稳定的匹配与最大的总满意率较低的配额。我们首先研究如何在四个自然场景中选择稳定匹配的总满意度的变化，并提供这些最大差距的确切值。随后，我们提出了一个策略证明的近似算法为我们的问题，在一种情况下，它最佳地解决了问题，并在其他三种情况下，这是NP难的，它产生了一个更好的近似因子比一个天真的平局打破方法。最后，我们展示了上述三个NP难场景的不可逼近性结果。
英文摘要：
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 because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings. In this paper, we formulate an optimization problem 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 and provide the exact values of these maximum gaps. 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 that of a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios.
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