Characterizing a Set of Popular Matchings Defined by Preference Lists with Ties
Tomomi Matsui, Takayoshi Hamaguchi
TL;DR
The paper addresses the problem of characterizing and computing popular matchings defined by preference lists with ties in a bipartite setting. It introduces a constructive characterization based on the first-choice graph $G_1$, a minimum $X$-cover of $G_1$, and the derived sets $\widetilde{P}$ and $\widetilde{E}$, proving that an applicant-complete matching is popular iff $M\subseteq \widetilde{E}$ and all posts in $\widetilde{P}$ are matched; popularity corresponds to optimality for a maximum-weight problem $MP$ on $G_2$, with a dual solution derived from the cover. This leads to a reduction of the minimum-cost popular matching problem to a minimum-cost assignment problem $MCP$ on $E_2$, solvable in $O(n(n\log n+m))$ time via successive shortest paths, after constructing $\widetilde{P}$ and $\widetilde{E}$ in $O(\sqrt{n}m)$ time. The work thus provides a complete, efficient framework for identifying the set of popular matchings with ties and for computing a minimum-cost member, linking to classical matching and duality theory. Practical impact includes enabling polynomial-time solutions for cost-constrained popular matchings in settings with ties and offering avenues for enumeration and faster variants for special structures.
Abstract
In this paper, we give a simple characterization of a set of popular matchings defined by preference lists with ties. By employing our characterization, we propose a polynomial time algorithm for finding a minimum cost popular matching.