Motivated by a serious issue that hospitals in rural areas suffer from shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the goal 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 famous rural hospitals theorem, so there is no room for algorithmic interventions. However, when ties are introduced in preference lists, this is not the case since the number of residents may vary over stable matchings. The main focus of this paper is to investigate how much we can utilize this flexibility to aid rural hospitals, in the presence of ties. We first show that the exact optimization is NP-hard and incompatible with strategy-proofness for residents. We then propose a linear-time strategy-proof algorithm whose approximation ratio is substantially better than a naive algorithm that breaks ties arbitrarily and applies the Gale-Shapley algorithm.
翻译:由于农村地区医院存在居民短缺的严重问题,我们研究了医院/居民模式,医院/居民模式与低配额挂钩,目标是尽可能满足这些模式。如果优惠名单严格,由于著名的农村医院的理论,分配给每个医院的居民人数在任何稳定匹配中都是相同的,因此没有进行算法干预的空间。但是,如果在优惠名单中引入联系,情况并非如此,因为居民人数可能因稳定匹配而不同。本文的主要焦点是调查我们如何利用这种灵活性帮助农村医院。我们首先显示,精确的优化是坚固的,与居民的战略防守不相容。我们然后提出一个线性战略防偏差的算法,其近似率比一个任意断绝关系并应用Gale-Shapley算法的天真算法要好得多。