We consider a matching problem in a bipartite graph $G$ where every vertex has a capacity and a strict preference order on its neighbors. Furthermore, there is a cost function on the edge set. We assume $G$ admits a perfect matching, i.e., one that fully matches all vertices. It is only perfect matchings that are feasible for us and we are interested in those perfect matchings that are popular within the set of perfect matchings. It is known that such matchings (called popular perfect matchings) always exist and can be efficiently computed. What we seek here is not any popular perfect matching, but a min-cost one. We show a polynomial-time algorithm for finding such a matching; this is via a characterization of popular perfect matchings in $G$ in terms of stable matchings in a colorful auxiliary instance. This is a generalization of such a characterization that was known in the one-to-one setting.

翻译：我们考虑一个二分图$G$中的匹配问题，其中每个顶点具有容量限制，并对其邻居持有严格的偏好顺序。此外，边集上存在一个成本函数。我们假设$G$允许完美匹配，即能够完全匹配所有顶点的匹配。仅完美匹配对我们而言是可行的，我们关注的是在完美匹配集合中具有流行性的那些完美匹配。已知此类匹配（称为流行完美匹配）始终存在且可高效计算。我们在此寻求的并非任意流行完美匹配，而是最小成本的流行完美匹配。我们提出一种多项式时间算法来寻找此类匹配；该算法通过将$G$中的流行完美匹配表征为一个彩色辅助实例中的稳定匹配来实现。这是一对一设定中已知表征方法的推广。