Let G = (A∪B, E) be a bipartite graph where the set A consists of agents or main players and the set B consists of jobs or secondary players. Every vertex in A∪B has a strict ranking of its neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M to N is at least the number that prefer N to M. Popular matchings always exist in G since every stable matching is popular. A matching M is A-popular if for any matching N, the number of agents (i.e., vertices in A) that prefer M to N is at least the number of agents that prefer N to M. Unlike popular matchings, A-popular matchings need not exist in a given instance G and there is a simple linear time algorithm to decide if G admits an A-popular matching and compute one, if so.

We consider the problem of deciding if G admits a matching that is both popular and A-popular and finding one, if so. We call such matchings fully popular. A fully popular matching is useful when A is the more important side—so along with overall popularity, we would like to maintain “popularity within the set A”. A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial-time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.

令G = ( A ∪ B , E ) 为二分图，其中集合A由代理人或主要参与者组成，集合B由工作或次要参与者组成。A ∪ B中的每个顶点对其邻居都有严格的排序。如果对于任何匹配N ，相比N更喜欢M 的顶点数量至少是比 M 更喜欢 N 的顶点数量，则匹配M是受欢迎的。热门匹配总是存在于G中，因为每个稳定的匹配都是热门的。如果对于任何匹配N ，相比N更喜欢M的智能体（即A中的顶点）的数量至少是比M更喜欢N 的智能体的数量，则匹配M是A流行的。与流行匹配不同，A流行匹配不需要存在于给定实例G中，并且有一个简单的线性时间算法来决定G是否承认A流行匹配并计算一个，如果是的话。

我们考虑这样的问题：决定G是否承认一个既受欢迎又A受欢迎的匹配，如果是的话，找到一个。我们称这种匹配为完全流行。当A是更重要的一方时，完全流行的匹配是有用的- 因此，除了整体流行度之外，我们还希望保持“  A组内的流行度”。完全流行的匹配不一定是最小尺寸/最大尺寸的流行匹配，并且所有已知的流行匹配问题的多项式时间算法都计算最小尺寸或最大尺寸的流行匹配。在这里，我们展示了完全流行的匹配问题的线性时间算法，因此我们的结果显示了流行匹配的新的易于处理的子类。