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 while obtaining NP-completeness if two instances differ only by a downward shift of one alternative by four agents. Moreover, we find a complexity dichotomy based on preference completeness for the case where instances differ by making some options unavailable.

翻译：我们研究了偏好下的匹配流行性。这一解概念捕捉了在代理人的多数投票中不输给任何其他匹配的匹配。如果一个流行匹配在多个实例中都是流行的，则称其为鲁棒流行匹配。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，而当两个实例因四个代理人对某个选项的向下移动而不同时，该问题被证明是NP完全的。此外，我们基于偏好完整性，对实例因某些选项不可用而不同的情况，发现了一个复杂性二分法。