Recently MV18 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:最近, MV18 查明并启动了关于理解两个“近亲”稳定匹配情况下两个“近亲”相匹配解决方案的拉特方之间结构关系新问题的工程。 它们还应用了自己的工作来找到一个稳稳定的匹配。 但是, 允许从美元到美元美元的新汇率的变更类型非常受限制, 即任何一个代理机构都会执行上调。 在本文中, 我们允许任何一个代理机构任意修正其偏爱列表。 允许任何一个代理商任意修正其偏爱列表。 允许任何代理商任意地 。 $M_ A$和$M_B$美元是由此产生的两个“ 近亲近” 解决方案中两个“ 近亲” 稳定匹配方之间结构关系的新问题 。 并且让 美元( 美元 美元 美元 ) 和 美元 ( 美元 美元 ) 和 美元 等两个“ 稳定匹配方 ” 。 让 美元和 美元 美元 和 美元 等的相对匹配组合一个“ ” ( 美元 ), ( 美元) ( 美元) ( 美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) ) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元)