Solving the Maximum Popular Matching Problem with Matroid Constraints

TL;DR

This paper introduces a polynomial-time algorithm for maximum popular matchings under matroid constraints in many-to-many settings, extending previous work limited to base orderable matroids, and explores complexity issues under lexicographic preferences.

Contribution

It proves the problem is tractable for arbitrary matroids using a new exchange property, and analyzes the complexity of popularity under lexicographic preferences.

Findings

Polynomial-time algorithm for arbitrary matroids.

NP-hardness results for lexicographic preferences.

Extension of popular matching theory to complex constraints.

Abstract

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 coNP-hard, even in the $b$-matching case.