Testing popularity in linear time via maximum matching

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics Pub Date : 2025-01-24 DOI:10.1016/j.dam.2025.01.014

Erika Bérczi-Kovács , Kata Kosztolányi

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引用次数: 0

Abstract

Popularity is an approach in mechanism design to find fair structures in a graph, based on the votes of the nodes. Popular matchings are the relaxation of stable matchings: given a graph

G = (V, E)

with strict preferences on the neighbors of the nodes, a matching

M

is popular if there is no other matching

M^{'}

such that the number of nodes preferring

M^{'}

is more than those preferring

M

. This paper considers the popularity testing problem, when the task is to decide whether a given matching is popular or not. Previous algorithms applied reductions to maximum weight matchings. We give a new algorithm for testing popularity by reducing the problem to maximum matching testing, thus attaining a linear running time

O (| E |)

Linear programming-based characterization of popularity is often applied for proving the popularity of a certain matching. As a consequence of our algorithm we derive a more structured dual witness than previous ones. Based on this result we give a combinatorial characterization of fractional popular matchings, which is a special class of popular matchings.

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来源期刊

Discrete Applied Mathematics 数学-应用数学

CiteScore

2.30

自引率

9.10%

发文量

422

审稿时长

4.5 months

期刊介绍： The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal. Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.