Testing popularity in linear time via maximum matching

IF 1 3区数学 Q3 MATHEMATICS, APPLIED Discrete Applied Mathematics Pub Date : 2025-05-15 Epub 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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通过最大匹配在线性时间内测试人气

人气是一种机制设计方法，基于节点的投票，在图中找到公平的结构。流行匹配是稳定匹配的松弛：给定一个对节点的邻居具有严格偏好的图G=(V，E)，如果不存在其他匹配M ‘，使得偏好M ’的节点数大于偏好M的节点数，则匹配M是流行的。本文考虑了流行度测试问题，其任务是确定给定匹配是否流行。以前的算法对最大权重匹配进行约简。通过将问题简化为最大匹配测试，给出了一种新的流行度测试算法，从而获得了线性运行时间O（|E|）。基于线性规划的流行度表征常用于证明某一匹配的流行度。作为我们算法的结果，我们得到了一个比以前更结构化的双重见证。在此基础上，我们给出了分数阶流行匹配的组合表征，分数阶流行匹配是一类特殊的流行匹配。

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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.

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