Solving the Maximum Popular Matching Problem with Matroid Constraints – new study by Gergely Csáji and co-authors

 

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 [Theoret. Comput. Sci., 809 (2020), pp. 265–276] 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 coNP-hard even in the -matching case.