[论文解读] A Minimax Perspective on Almost-Stable Matchings

Frederik Glitzner, David F. Manlove|arXiv (Cornell University)|Jan 20, 2026

Game Theory and Voting Systems被引用 0

一句话总结

本文引入匹配问题的极小极大近似稳定性，给出强硬的NP-hardness结果，并在受限实例上提供积极的算法，同时通过经验验证表明存在较好分布的不稳定性。

ABSTRACT

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.

研究动机与目标

• 在无法实现完全稳定且不稳定性必须被公平分配的匹配市场中，激励对稳定性的研究。
• 引入以公平为导向的极小极大目标，目标是将任意代理所经历的阻塞对的最大数量降至最小。
• 表征极小极大近似稳定性在稳定匹配和同伴设置中的计算复杂性。
• 为偏好长度受限的情况提供积极的算法结果，并为一般情况开发近似和整数规划。
• 通过实证评估极小极大近似稳定匹配的存在性及质量。

提出的方法

• 将 Minimax-AlmostStable-sri 与 Minimax-AlmostStable-Max-smi 作为新的优化问题定义，聚焦于每个代理的最差阻塞对数量。
• 证明在 sri 与 smi 设置下，即使列表有边界或完全列表，决策变体也为 NP-完全。
• 给出偏好列表长度最多为 2 的实例的多项式时间算法，并为一般情况提供近似算法与整数规划的表达式。
• 将极小极大近似稳定性与现有的聚合不稳定性度量和阻塞代理数量进行对比，以建立结构性差异。
• 构建理论下界，展示极小极大值如何随问题规模增长，并在某些构造中给出紧性证明。

Figure 1 . Schematic illustration of the sri lower-bound construction for 27 agents (i.e,. $k=3$ ). Agents are partitioned into 3 nested sets containing $3^{j}$ (for $1\leq i\leq k$ ) agents each. Arrows indicate preference cycles within and between the nested levels.

实验结果

研究问题

• RQ1 在 sri 和 smi 设置下，找到极小极大近似稳定匹配的计算复杂性是什么？
• RQ2 对于受限偏好长度的情况，极小极大近似稳定匹配能否在多项式时间内计算？
• RQ3 在公平性与可行性方面，极小极大近似稳定匹配与聚合稳定性度量及阻塞代理计数相比有何差异？
• RQ4 在一般设置下，有哪些近似性与精确优化的方法可以解决极小极大近似稳定性问题？
• RQ5 实证分析是否表明实践中存在分布较好的不稳定性匹配？

主要发现

• 对于 sri 与 smi，判断是否存在每个代理至多一个阻塞对的匹配是 NP-完全的，即使列表长度是常数-bound。
• Minimax-AlmostStable-Max-smi 的最坏情况最优值为 OPT = Θ(n)，表明随代理数量线性增长。
• 存在偏好列表长度不超过 2 的实例的多项式时间算法。
• 本文提供了一般设置的近似算法与精确的整数规划模型。
• 实验表明存在具有良好不稳定性分布的几乎稳定与最大基数匹配。

Figure 2 . Illustration of the lower-bound construction for maximum-cardinality matchings in smi .