Stability is crucial in matching markets, yet in many real-world settings - from hospital residency allocations to roommate assignments - full stability is either impossible to achieve or can come at the cost of leaving many agents unmatched. When stability cannot be achieved, algorithmicists and market designers face a critical question: how should instability be measured and distributed among participants? Existing approaches to "almost-stable" matchings focus on aggregate measures, minimising either the total number of blocking pairs or the count of agents involved in blocking pairs. However, such aggregate objectives can result in concentrated instability on a few individual agents, raising concerns about fairness and incentives to deviate. We introduce a fairness-oriented approach to approximate stability based on the minimax principle: we seek matchings that minimise the maximum number of blocking pairs any agent is in. Equivalently, we minimise the maximum number of agents that anyone has justified envy towards. This distributional objective protects the worst-off agents from a disproportionate amount of instability. We characterise the computational complexity of this notion across fundamental matching settings. Surprisingly, even very modest guarantees prove computationally intractable: we show that it is NP-complete to decide whether a matching exists in which no agent is in more than one blocking pair, even when preference lists have constant-bounded length. This hardness applies to both Stable Roommates and maximum-cardinality Stable Marriage. On the positive side, we provide polynomial-time algorithms when agents rank at most two others, and present approximation algorithms and integer programs. Our results map the algorithmic landscape and reveal fundamental trade-offs between distributional guarantees and computational feasibility.

翻译：稳定性在匹配市场中至关重要，然而在许多现实场景中——从医院住院医师分配到室友匹配——完全稳定性要么无法实现，要么可能以牺牲大量参与者无法匹配为代价。当稳定性无法实现时，算法设计者与市场设计者面临一个关键问题：应如何衡量不稳定性并在参与者之间分配？现有“近似稳定”匹配方法主要关注聚合指标，即最小化阻塞对总数或涉及阻塞对的参与者数量。然而，此类聚合目标可能导致不稳定性集中在少数个体参与者身上，引发对公平性与偏离动机的担忧。我们基于最小化最大原则提出一种面向公平的近似稳定性方法：寻求最小化任一参与者所处阻塞对数量的最大值。等价而言，我们最小化任何人对其他参与者产生合理嫉妒的最大人数。这种分布性目标保护处境最差参与者免受过度的不稳定性影响。我们在基础匹配设定中刻画了这一概念的计算复杂性。令人惊讶的是，即使非常有限的保证也被证明是计算难解的：我们证明当偏好列表长度有常数界时，判定是否存在任一参与者至多处于一个阻塞对的匹配是NP完全问题。这一硬度结果同时适用于稳定室友问题与最大基数稳定婚姻问题。在积极方面，我们为参与者最多排序两个对象的场景提供多项式时间算法，并提出近似算法与整数规划方案。我们的研究结果描绘了算法版图，揭示了分布性保证与计算可行性之间的根本权衡。