We study the Popular Matching problem in multiple models, where the preferences of the agents in the instance may change or may be unknown/uncertain. In particular, we study an Uncertainty model, where each agent has a possible set of preferences, a Multilayer model, where there are layers of preference profiles, a Robust model, where any agent may move some other agents up or down some places in his preference list and an Aggregated Preference model, where votes are summed over multiple instances with different preferences. We study both one-sided and two-sided preferences in bipartite graphs. In the one-sided model, we show that all our problems can be solved in polynomial time by utilizing the structure of popular matchings. We also obtain nice structural results. With two-sided preferences, we show that all four above models lead to NP-hard questions for popular matchings. By utilizing the connection between dominant matchings and stable matchings, we show that in the robust and uncertainty model, a certainly dominant matching in all possible prefernce profiles can be found in polynomial-time, whereas in the multilayer and aggregated models, the problem remains NP-hard for dominant matchings too. We also answer an open question about $d$-robust stable matchings.

翻译：本文研究了多种模型下的流行匹配问题，其中实例中代理的偏好可能发生变化，或未知/不确定。具体而言，我们研究了不确定性模型（每个代理有一组可能的偏好）、多层模型（存在多层偏好轮廓）、鲁棒模型（任意代理可将其偏好列表中某些代理的位置上调或下调若干位）以及聚合偏好模型（对不同偏好实例的投票结果进行累加）。我们在二分图中同时研究了单侧偏好和双侧偏好两种情形。对于单侧偏好模型，我们证明通过利用流行匹配的结构，所有问题均可在多项式时间内求解，并获得了优美的结构性结论。对于双侧偏好模型，我们证明上述四种模型均导致流行匹配问题的NP难性。通过利用主导匹配与稳定匹配之间的关联，我们证明在鲁棒模型和不确定性模型中，可在多项式时间内找到所有可能偏好轮廓下确定性的主导匹配，而在多层模型和聚合模型中，主导匹配问题仍保持NP难性。此外，我们还解答了一个关于$d$-鲁棒稳定匹配的开放性问题。