We investigate weighted settings of popular matching problems with matroid constraints. The concept of popularity was originally defined for matchings in bipartite graphs, where vertices have preferences over the incident edges. There are two standard models depending on whether vertices on one or both sides have preferences. A matching $M$ is popular if it does not lose a head-to-head election against any other matching. In our generalized models, one or both sides have matroid constraints, and a weight function is defined on the ground set. Our objective is to find a popular optimal matching, i.e., a maximum-weight matching that is popular among all maximum-weight matchings satisfying the matroid constraints. For both one- and two-sided preferences models, we provide efficient algorithms to find such solutions, combining algorithms for unweighted models with fundamental techniques from combinatorial optimization. The algorithm for the one-sided preferences model is further extended to a model where the weight function is generalized to an M$^\natural$-concave utility function. Finally, we complement these tractability results by providing hardness results for the problems of finding a popular near-optimal matching. These hardness results hold even without matroid constraints and with very restricted weight functions.

翻译：本文研究了带拟阵约束的加权流行匹配问题。流行性概念最初针对二部图匹配提出，其中顶点对关联边存在偏好。根据单侧或双侧顶点具有偏好，存在两种标准模型。若匹配$M$在与任何其他匹配的"一对一"选举中均不落败，则称其为流行匹配。在我们的广义模型中，单侧或双侧存在拟阵约束，并在基础集上定义了权重函数。我们的目标是寻找流行最优匹配，即在满足拟阵约束的所有最大权重匹配中具有流行性的匹配。针对单侧与双侧偏好模型，我们结合非加权模型算法与组合优化的基本技术，给出了求解此类匹配的高效算法。针对单侧偏好模型的算法可进一步推广至权重函数扩展为M$^\natural$凹效用函数的模型。最后，我们通过证明寻找流行近似最优匹配问题的困难性结果，完善了这些可解性结论。这些困难性结果即使在没有拟阵约束且权重函数高度受限的情况下依然成立。