Popular Maximum-Utility Matchings with Matroid Constraints

Gergely Csáji, Tamás Király, Kenjiro Takazawa, Yu Yokoi
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Abstract

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.
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带 Matroid 约束的流行最大效用匹配
我们研究了具有矩阵约束的流行匹配问题的加权设置。流行度的概念最初是为两方图中的匹配定义的,其中顶点对入射边有偏好。如果一个匹配 $M$ 在与其他匹配的竞争中没有失利,那么它就是受欢迎的。在我们的广义模型中,一方或双方都有矩阵约束,并且在地面集上定义了权重函数。我们的目标是找到一个受欢迎的最优匹配,即在所有满足矩阵约束条件的最大权重匹配中受欢迎的最大权重匹配。对于单边和双边偏好模型,我们都提供了高效的算法来找到这样的解,并将非加权模型的算法与组合优化的基本技术相结合。单边偏好模型的算法进一步扩展到了权重函数泛化为 M$^\natural$-concave 效用函数的模型。最后,我们对可计算性结果进行了补充,提供了寻找流行的近优匹配问题的硬度结果。即使没有机器人约束和非常有限的权重函数,这些硬度结果也是成立的。
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