Stable matching is a fundamental area with many practical applications, such as centralised clearinghouses for school choice or job markets. Recent work has introduced the paradigm of near-feasibility in capacitated matching settings, where agent capacities are slightly modified to ensure the existence of desirable outcomes. While useful when no stable matching exists, or some agents are left unmatched, it has not previously been investigated whether near-feasible stable matchings satisfy desirable properties with regard to their stability in the original instance. Furthermore, prior works often leave open deviation incentive issues that arise when the centralised authority modifies agents' capacities. We consider these issues in the Stable Fixtures problem model, which generalises many classical models through non-bipartite preferences and capacitated agents. We develop a formal framework to analyse and quantify agent incentives to adhere to computed matchings. Then, we embed near-feasible stable matchings in this framework and study the trade-offs between instability, capacity modifications, and computational complexity. We prove that capacity modifications can be simultaneously optimal at individual and aggregate levels, and provide efficient algorithms to compute them. We show that different modification strategies significantly affect stability, and establish that minimal modifications and minimal deviation incentives are compatible and efficiently computable under general conditions. Finally, we provide exact algorithms and experimental results for tractable and intractable versions of these problems.

翻译：稳定匹配是一个具有众多实际应用的基础领域，例如学校选择或就业市场的集中清算机制。近期研究在容量匹配场景中引入了近似可行性范式，即通过微调代理容量来确保理想结果的存在。尽管在不存在稳定匹配或部分代理无法匹配时该方法具有实用性，但此前尚未探究近似可行稳定匹配是否在原实例的稳定性方面满足理想特性。此外，先前研究往往未解决集中管理机构调整代理容量时产生的偏离激励问题。我们在稳定配置问题模型（通过非二分偏好和容量化代理泛化了诸多经典模型）中探讨这些问题。我们建立了形式化框架来分析与量化代理遵守计算匹配的激励程度，进而将近似可行稳定匹配嵌入该框架，研究不稳定性、容量调整与计算复杂度之间的权衡关系。我们证明容量调整可在个体与聚合层面同时达到最优，并提供高效计算算法。研究表明不同调整策略会显著影响稳定性，并论证在一般条件下最小化调整与最小化偏离激励具有兼容性且可高效计算。最后，我们针对这些问题的可处理与难处理版本分别给出精确算法与实验结果。