We study stable matchings that are robust to preference changes in the two-sided stable matching setting of Gale and Shapley [GS62]. Given two instances $A$ and $B$ on the same set of agents, a matching is said to be robust if it is stable under both instances. This notion captures desirable robustness properties in matching markets where preferences may evolve, be misreported, or be subject to uncertainty. While the classical theory of stable matchings reveals rich lattice, algorithmic, and polyhedral structure for a single instance, it is unclear which of these properties persist when stability is required across multiple instances. Our work initiates a systematic study of the structural and computational behavior of robust stable matchings under increasingly general models of preference changes. We analyze robustness under a hierarchy of perturbation models: 1. a single upward shift in one agent's preference list, 2. an arbitrary permutation change by a single agent, and 3. arbitrary preference changes by multiple agents on both sides. For each regime, we characterize when: 1. the set of robust stable matchings forms a sublattice, 2. the lattice of robust stable matchings admits a succinct Birkhoff partial order enabling efficient enumeration, 3. worker-optimal and firm-optimal robust stable matchings can be computed efficiently, and 4. the robust stable matching polytope is integral (by studying its LP formulation). We provide explicit counterexamples demonstrating where these structural and geometric properties break down, and complement these results with XP-time algorithms running in $O(n^k)$ time, parameterized by $k$, the number of agents whose preferences change. Our results precisely delineate the boundary between tractable and intractable cases for robust stable matchings.

翻译：我们在Gale和Shapley [GS62]提出的双边稳定匹配框架下，研究对偏好变化具有鲁棒性的稳定匹配。给定同一组智能体上的两个实例$A$和$B$，若某个匹配在两个实例下均保持稳定，则称其为鲁棒匹配。这一概念捕捉了匹配市场中理想的鲁棒性特征，因为市场中的偏好可能随时间演变、被误报或存在不确定性。虽然经典稳定匹配理论针对单一实例揭示了丰富的格结构、算法特性及多面体结构，但当要求匹配在多个实例间均保持稳定时，尚不清楚这些性质中有哪些能够得以保留。本研究首次系统性地探讨了在日益普遍的偏好变化模型下，鲁棒稳定匹配的结构特性与计算行为。我们基于以下递进的扰动模型层次分析鲁棒性：1. 单个智能体偏好列表中的一次向上移位；2. 单个智能体的任意排列变化；3. 双边多个智能体的任意偏好变化。针对每种情形，我们刻画了以下特征：1. 鲁棒稳定匹配集合何时构成子格；2. 鲁棒稳定匹配格何时具有简洁的Birkhoff偏序以实现高效枚举；3. 工人最优与厂商最优鲁棒稳定匹配何时可被高效计算；4. 鲁棒稳定匹配多面体何时具有完整性（通过研究其线性规划表述）。我们提供了明确的反例以展示这些结构性质与几何特性在何处失效，并辅以参数化XP时间算法作为补充，这些算法的运行时间为$O(n^k)$，其中参数$k$表示偏好发生变化的智能体数量。我们的研究结果精确界定了鲁棒稳定匹配问题可处理与难处理情形之间的边界。