Popularity and Perfectness in One-sided Matching Markets with Capacities
Gergely Csáji
TL;DR
The paper investigates many-to-one house allocation with capacities, focusing on one-sided preferences. It proves NP-hardness for recognizing popular matchings when applicants have capacities, and then analyzes capacity-adjustment strategies on the house side to obtain perfect and either Pareto-optimal or popular matchings, under Min-Sum and Min-Max objectives. A central contribution is a polynomial-time algorithm for Min-Sumpop-p when only increases are allowed, built on a structural G'-based characterization with admirers, while Min-Max variants are NP-hard and hard to approximate. The results reveal a nuanced landscape: increasing capacities can yield efficient solutions for popularity under certain criteria, but allowing decreases or optimizing the maximum change generally leads to intractability. The work also contrasts with two-sided preference results and contributes a precise operational framework (admirers, G', and capacity-change lemmas) for capacity design in one-sided settings.
Abstract
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.