We consider many-to-one matching problems, where one side corresponds to applicants who have preferences and the other side to houses who do not have preferences. We consider two different types of this market: one, where the applicants have capacities, and one where the houses do. First, we answer an open question by Manlove and Sng (2006) (partly solved Paluch (2014) for preferences with ties), that is, we show that deciding if a popular matching exists in the house allocation problem, where agents have capacities is NP-hard for previously studied versions of popularity. Then, we consider the other version, where the houses have capacities. We study how to optimally increase the capacities of the houses to obtain a matching satisfying multiple optimality criteria, like popularity, Pareto-optimality and perfectness. We consider two common optimality criteria, one aiming to minimize the sum of capacity increases of all houses and the other aiming to minimize the maximum capacity increase of any school. We obtain a complete picture in terms of computational complexity and some algorithms.

翻译：我们研究多对一的匹配问题，其中一方对应具有偏好的申请者，另一方则对应无偏好的房屋。我们考虑该市场的两类不同情形：一是申请者受容量限制，二是房屋受容量限制。首先，我们解决了Manlove与Sng（2006）提出的开放性问题（Paluch（2014）针对含平局偏好情形已部分解决），即证明在房屋分配问题中，当代理人具有容量限制时，判断是否存在流行匹配对于此前研究的流行性定义而言是NP难的。随后，我们考察房屋存在容量限制的另一类情形，研究如何最优地增加房屋容量以获得满足多重最优性准则（如流行性、帕累托最优性及完美性）的匹配。我们采用两个常见的最优性准则：一者旨在最小化所有房屋容量增加的总和，另一者则旨在最小化任意学校容量增加的最大值。我们完整刻画了相应问题的计算复杂度，并提出了若干算法。