{"title":"Popular Matchings with One-Sided Bias","authors":"Telikepalli Kavitha","doi":"10.1145/3638764","DOIUrl":null,"url":null,"abstract":"<p>Let <i>G</i> = (<i>A</i>∪<i>B</i>, <i>E</i>) be a bipartite graph where the set <i>A</i> consists of agents or main players and the set <i>B</i> consists of jobs or secondary players. Every vertex in <i>A</i>∪<i>B</i> has a strict ranking of its neighbors. A matching <i>M</i> is <i>popular</i> if for any matching <i>N</i>, the number of vertices that prefer <i>M</i> to <i>N</i> is at least the number that prefer <i>N</i> to <i>M</i>. Popular matchings always exist in <i>G</i> since every stable matching is popular. A matching <i>M</i> is <i><i>A</i>-popular</i> if for any matching <i>N</i>, the number of <i>agents</i> (i.e., vertices in <i>A</i>) that prefer <i>M</i> to <i>N</i> is at least the number of agents that prefer <i>N</i> to <i>M</i>. Unlike popular matchings, <i>A</i>-popular matchings need not exist in a given instance <i>G</i> and there is a simple linear time algorithm to decide if <i>G</i> admits an <i>A</i>-popular matching and compute one, if so. </p><p>We consider the problem of deciding if <i>G</i> admits a matching that is both popular and <i>A</i>-popular and finding one, if so. We call such matchings <i>fully popular</i>. A fully popular matching is useful when <i>A</i> is the more important side—so along with overall popularity, we would like to maintain “popularity within the set <i>A</i>”. A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial-time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"462 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Algorithms","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1145/3638764","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
Let G = (A∪B, E) be a bipartite graph where the set A consists of agents or main players and the set B consists of jobs or secondary players. Every vertex in A∪B has a strict ranking of its neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M to N is at least the number that prefer N to M. Popular matchings always exist in G since every stable matching is popular. A matching M is A-popular if for any matching N, the number of agents (i.e., vertices in A) that prefer M to N is at least the number of agents that prefer N to M. Unlike popular matchings, A-popular matchings need not exist in a given instance G and there is a simple linear time algorithm to decide if G admits an A-popular matching and compute one, if so.
We consider the problem of deciding if G admits a matching that is both popular and A-popular and finding one, if so. We call such matchings fully popular. A fully popular matching is useful when A is the more important side—so along with overall popularity, we would like to maintain “popularity within the set A”. A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial-time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.
假设 G = (A∪B, E) 是一个双方图，其中集合 A 由代理或主要参与者组成，集合 B 由工作或次要参与者组成。A∪B 中的每个顶点都有一个严格的邻接排序。如果对任意匹配 N 而言，喜欢 M 多于 N 的顶点数至少等于喜欢 N 多于 M 的顶点数，那么匹配 M 就是受欢迎的。与流行匹配不同，A-流行匹配不一定存在于给定的实例 G 中，而且有一种简单的线性时间算法来判断 G 是否允许 A-流行匹配，如果允许，则计算一个 A-流行匹配。我们考虑的问题是判断 G 是否存在既流行又 A 流行的匹配，如果存在，则找出一个。我们称这种匹配为完全流行匹配。当 A 是更重要的一方时，完全受欢迎的匹配是有用的--因此除了整体受欢迎程度，我们还希望保持 "集合 A 中的受欢迎程度"。完全流行匹配不一定是最小/最大流行匹配，所有已知的流行匹配问题多项式时间算法计算的都是最小或最大流行匹配。在这里，我们展示了完全流行匹配问题的线性时间算法，因此我们的结果显示了流行匹配的一个新的可操作性子类。
期刊介绍：
ACM Transactions on Algorithms welcomes submissions of original research of the highest quality dealing with algorithms that are inherently discrete and finite, and having mathematical content in a natural way, either in the objective or in the analysis. Most welcome are new algorithms and data structures, new and improved analyses, and complexity results. Specific areas of computation covered by the journal include
combinatorial searches and objects;
counting;
discrete optimization and approximation;
randomization and quantum computation;
parallel and distributed computation;
algorithms for
graphs,
geometry,
arithmetic,
number theory,
strings;
on-line analysis;
cryptography;
coding;
data compression;
learning algorithms;
methods of algorithmic analysis;
discrete algorithms for application areas such as
biology,
economics,
game theory,
communication,
computer systems and architecture,
hardware design,
scientific computing
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