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. Except for the Robust model, these hardness results hold even if one side of the agents has always the same preferences.

翻译：我们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，或存在未知/不确定性。具体而言，我们考察了以下模型：不确定性模型（每个代理具有一组可能的偏好）、多层次模型（存在多层偏好概况）、鲁棒模型（任意代理可将其偏好列表中其他代理的位置上下移动若干位次）以及聚合偏好模型（对不同偏好实例上的投票结果进行加总）。我们同时研究了二分图中单边与双边偏好情形。在单边偏好模型中，我们证明所有问题均可通过利用流行匹配的结构在多项式时间内求解，并获得了优美的结构性结论。而在双边偏好模型下，我们发现上述四种模型均导致NP难题。除鲁棒模型外，即便代理单边拥有固定偏好，这些困难性结论依然成立。