带拟阵约束的流行最大效用匹配
Popular Maximum-Utility Matchings with Matroid Constraints
AI总结：
针对带拟阵约束的加权流行匹配问题，提出求解单侧/双侧偏好模型下流行最优匹配的高效算法，并扩展至M$^\ atural$-凹效用场景，同时给出流行近最优匹配问题的难度结论。
摘要：
我们研究了带拟阵约束的流行匹配问题的加权场景。“流行性”这一概念最初是针对二分图中的匹配定义的，其中顶点对关联边具有偏好。根据是单侧还是双侧顶点具有偏好，存在两种标准模型。若一个匹配$M$在与其他任何匹配的一对一投票中都不会落败，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，且基础集上定义了权重函数。我们的目标是找到流行最优匹配，即在所有满足拟阵约束的最大权重匹配中具备流行性的最大权重匹配。针对单侧偏好和双侧偏好两种模型，我们结合无权重模型算法与组合优化的基础技术，给出了求解此类解的高效算法。针对单侧偏好模型的算法被进一步扩展到权重函数被推广为M$^\ atural$-凹效用函数的模型中。最后，我们通过给出寻找流行近最优匹配问题的难度结果，补充了这些可解性结论。这些难度结果甚至在无拟阵约束且权重函数受限性极强的情况下依然成立。
英文摘要：
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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