We study popularity for matchings under preferences. This solution concept captures matchings that do not lose against any other matching in a majority vote by the agents. A popular matching is said to be robust if it is popular among multiple instances. We present a polynomial-time algorithm for deciding whether there exists a robust popular matching if instances only differ with respect to the preferences of a single agent. The same method applies also to dominant matchings, a subclass of maximum-size popular matchings. By contrast, we obtain NP-completeness if two instances differ only by two agents of the same side or by a swap of two adjacent alternatives by two agents. The first hardness result applies to dominant matchings as well. Moreover, we find another complexity dichotomy based on preference completeness for the case where instances differ by making some options unavailable. We conclude by discussing related models, such as strong and mixed popularity.

翻译：本研究探讨偏好下的匹配流行性问题。该解概念刻画了在代理人的多数投票中不会输给任何其他匹配的匹配。若一个流行匹配在多个实例中均保持流行性，则称其为鲁棒的。我们提出了一种多项式时间算法，用于判断当实例仅因单个代理人的偏好不同时是否存在鲁棒流行匹配。该方法同样适用于支配匹配——最大规模流行匹配的一个子类。相比之下，若两个实例仅因同侧两个代理人的偏好差异，或两个代理人对相邻选项的交换而产生差异，则该问题被证明是NP完全的。首个硬度结论同样适用于支配匹配。此外，针对因部分选项不可用而产生实例差异的情形，我们基于偏好完备性发现了另一复杂度二分现象。最后，我们讨论了相关模型，如强流行性与混合流行性。