We consider many-to-one matching problems, where one side consists of students and the other side of schools with capacity constraints. We study how to optimally increase the capacities of the schools so as to obtain a stable and perfect matching (i.e., every student is matched) or a matching that is stable and Pareto-efficient for the students. We consider two common optimality criteria, one aiming to minimize the sum of capacity increases of all schools (abbrv. as MinSum) and the other aiming to minimize the maximum capacity increase of any school (abbrv. as MinMax). We obtain a complete picture in terms of computational complexity: Except for stable and perfect matchings using the MinMax criteria which is polynomial-time solvable, all three remaining problems are NP-hard. We further investigate the parameterized complexity and approximability and find that achieving stable and Pareto-efficient matchings via minimal capacity increases is much harder than achieving stable and perfect matchings.

翻译：我们考虑多对一匹配问题，其中一方由学生组成，另一方为具有容量约束的学校。我们研究如何最优地增加学校容量，以获得稳定且完美的匹配（即每名学生均被匹配）或对学生而言稳定且帕累托有效的匹配。我们考虑两种常见的最优准则：其一旨在最小化所有学校的容量增加总和（简称MinSum），另一旨在最小化任意学校的最大容量增加（简称MinMax）。我们在计算复杂性方面获得了完整图景：除基于MinMax准则的稳定完美匹配可在多项式时间内求解外，其余三个问题均为NP难问题。我们进一步研究参数化复杂性与可近似性，发现通过最小容量增加实现稳定且帕累托有效的匹配，远比实现稳定完美匹配困难。