Popularity and Perfectness in One-sided Matching Markets with Capacities

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.