We consider the problem of finding a maximum popular matching in a many-to-many matching setting with two-sided preferences and matroid constraints. This problem was proposed by Kamiyama (2020) and solved in the special case where matroids are base orderable. Utilizing a newly shown matroid exchange property, we show that the problem is tractable for arbitrary matroids. We further investigate a different notion of popularity, where the agents vote with respect to lexicographic preferences, and show that both existence and verification problems become NP-hard, even in the $b$-matching case.

翻译：我们研究在双边偏好和拟阵约束下，寻找多对多匹配场景中最大流行匹配的问题。该问题由Kamiyama（2020）提出，并在拟阵为基可排序的特殊情形下得到解决。利用新证明的拟阵交换性质，我们证明该问题对任意拟阵都是可处理的。我们进一步研究了另一种流行度概念，即代理根据字典序偏好进行投票，并证明即使在$b$-匹配情形下，存在性问题和验证问题均变为NP困难。