We study the problem of capacity modification in the many-to-one stable matching of workers and firms. Our goal is to systematically study how the set of stable matchings changes when some seats are added to or removed from the firms. We make three main contributions: First, we examine whether firms and workers can improve or worsen upon changing the capacities under worker-proposing and firm-proposing deferred acceptance algorithms. Second, we study the computational problem of adding or removing seats to either match a fixed worker-firm pair in some stable matching or make a fixed matching stable with respect to the modified problem. We develop polynomial-time algorithms for these problems when only the overall change in the firms' capacities is restricted, and show NP-hardness when there are additional constraints for individual firms. Lastly, we compare capacity modification with the classical model of preference manipulation by firms and identify scenarios under which one mode of manipulation outperforms the other. We find that a threshold on a given firm's capacity, which we call its peak, crucially determines the effectiveness of different manipulation actions.

翻译：我们研究了工人与企业多对一稳定匹配中的容量调整问题。目标在于系统性地探究当企业增加或移除部分岗位时，稳定匹配集合如何发生变化。本文主要贡献有三：首先，研究了在工人提议与企业提议的延迟接受算法下，变更容量是否会导致企业或工人的状况改善或恶化。其次，研究了通过增删岗位使某个固定的工人-企业对在某个稳定匹配中成功匹配，或者使某个固定匹配在调整后问题中成为稳定匹配的计算问题。当仅限制企业总容量变化时，我们为这些问题设计了多项式时间算法；当存在针对单个企业的额外约束时，证明其为NP困难问题。最后，将容量调整与经典的企业偏好操纵模型进行对比，并识别出某种操纵模式优于另一种模式的情景。研究发现，企业容量存在一个阈值（称为其峰值），该阈值对各类操纵行为的有效性具有决定性影响。