Characterizing a Set of Popular Matchings Defined by Preference Lists with Ties

2016-01-14

arXiv.org

AI Summary

This paper studies the popular matching problem under preference lists with ties, aiming to provide a concise combinatorial characterization of the set of popular matchings and to compute a minimum-cost popular matching. Addressing the limitation of prior theory—which only applies to strict preferences—we establish, for the first time, a necessary and sufficient combinatorial characterization of popular matchings in the presence of ties. Leveraging this characterization, we formulate a graph-theoretic model and design a polynomial-time algorithm that computes a minimum-cost popular matching in $O(n^3)$ time. Our work significantly extends the theoretical scope of popular matchings beyond strict preferences and provides an efficiently computable solution for an important generalization of stable matching. It thus advances both the theoretical foundations—by unifying and generalizing key structural insights—and practical applicability—by enabling scalable computation in real-world settings where indifference is prevalent.

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.

Problem

• Characterize popular matchings with tied preferences
• Develop polynomial-time algorithm for minimum cost
• Simplify matching problem using preference lists

Innovation

• Characterizes popular matchings with ties
• Proposes polynomial time algorithm
• Finds minimum cost popular matching