Recently MV18a identified and initiated work on the new problem of understanding structural relationships between the lattices of solutions of two ``nearby'' instances of stable matching. They also gave an application of their work to finding a robust stable matching. However, the types of changes they allowed in going from instance $A$ to $B$ were very restricted, namely, any one agent executes an upward shift. In this paper, we allow any one agent to permute its preference list arbitrarily. Let $M_A$ and $M_B$ be the sets of stable matchings of the resulting pair of instances $A$ and $B$, and let $\mathcal{L}_A$ and $\mathcal{L}_B$ be the corresponding lattices of stable matchings. We prove that the matchings in $M_A \cap M_B$ form a sublattice of both $\mathcal{L}_A$ and $\mathcal{L}_B$ and those in $M_A \setminus M_B$ form a join semi-sublattice of $\mathcal{L}_A$. These properties enable us to obtain a polynomial time algorithm for not only finding a stable matching in $M_A \cap M_B$, but also for obtaining the partial order, as promised by Birkhoff's Representation Theorem, thereby enabling us to generate all matchings in this sublattice. Our algorithm also helps solve a version of the robust stable matching problem. We discuss another potential application, namely obtaining new insights into the incentive compatibility properties of the Gale-Shapley Deferred Acceptance Algorithm.

翻译：近期MV18a提出并初步研究了关于两个“相邻”稳定匹配实例的解格之间结构关系的新问题，同时展示了该工作在一类鲁棒稳定匹配求解中的应用。然而，他们在从实例A到B的转化过程中所允许的变更类型相当受限——仅限于任意代理执行向上移位。本文允许任意代理任意排列其偏好列表。设M_A和M_B分别为实例对A与B的稳定匹配集合，L_A与L_B为相应的稳定匹配格。我们证明：M_A∩M_B中的匹配同时构成L_A与L_B的子格，而M_A\M_B中的匹配则构成L_A的并半子格。这些性质使我们能够获得多项式时间算法，不仅可用于寻找M_A∩B中的稳定匹配，还能依据Birkhoff表示定理获取偏序结构，进而生成该子格中的所有匹配。该算法同时有助于求解一类鲁棒稳定匹配问题。我们另探讨了另一潜在应用方向：即对Gale-Shapley延迟接受算法的激励相容性质提供新见解。