{"title":"受欢迎的室友在指数时间","authors":"Telikepalli Kavitha","doi":"10.1007/s00453-024-01287-4","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the popular matching problem in a <i>roommates</i> instance <i>G</i> on <i>n</i> vertices, i.e., <i>G</i> is a graph where each vertex has a strict preference order over its neighbors. A matching <i>M</i> is <i>popular</i> if there is no matching <i>N</i> such that the vertices that prefer <i>N</i> to <i>M</i> outnumber those that prefer <i>M</i> to <i>N</i>. It is known that it is NP-hard to decide if <i>G</i> admits a popular matching or not. There is no better algorithm known for this problem than the brute force algorithm that enumerates all matchings and tests each for popularity—this could take <i>n</i>! time. Here we show an <span>\\(O^*(k^n)\\)</span> time algorithm for this problem, where <span>\\(k < 7.32\\)</span>. We use the recent breakthrough result on the maximum number of stable matchings possible in a roommates instance to analyze our algorithm for the popular matching problem. We identify a natural (also, hard) subclass of popular matchings called <i>truly popular</i> matchings that are “popular fractional” and show an <span>\\(O^*(2^n)\\)</span> time algorithm for the truly popular matching problem in <i>G</i>. We also identify a subclass of max-size popular matchings called <i>super-dominant</i> matchings and show a linear time algorithm for the super-dominant roommates problem.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 2","pages":"292 - 320"},"PeriodicalIF":0.9000,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01287-4.pdf","citationCount":"0","resultStr":"{\"title\":\"Popular Roommates in Simply Exponential Time\",\"authors\":\"Telikepalli Kavitha\",\"doi\":\"10.1007/s00453-024-01287-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider the popular matching problem in a <i>roommates</i> instance <i>G</i> on <i>n</i> vertices, i.e., <i>G</i> is a graph where each vertex has a strict preference order over its neighbors. A matching <i>M</i> is <i>popular</i> if there is no matching <i>N</i> such that the vertices that prefer <i>N</i> to <i>M</i> outnumber those that prefer <i>M</i> to <i>N</i>. It is known that it is NP-hard to decide if <i>G</i> admits a popular matching or not. There is no better algorithm known for this problem than the brute force algorithm that enumerates all matchings and tests each for popularity—this could take <i>n</i>! time. Here we show an <span>\\\\(O^*(k^n)\\\\)</span> time algorithm for this problem, where <span>\\\\(k < 7.32\\\\)</span>. We use the recent breakthrough result on the maximum number of stable matchings possible in a roommates instance to analyze our algorithm for the popular matching problem. We identify a natural (also, hard) subclass of popular matchings called <i>truly popular</i> matchings that are “popular fractional” and show an <span>\\\\(O^*(2^n)\\\\)</span> time algorithm for the truly popular matching problem in <i>G</i>. We also identify a subclass of max-size popular matchings called <i>super-dominant</i> matchings and show a linear time algorithm for the super-dominant roommates problem.</p></div>\",\"PeriodicalId\":50824,\"journal\":{\"name\":\"Algorithmica\",\"volume\":\"87 2\",\"pages\":\"292 - 320\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-12-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00453-024-01287-4.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algorithmica\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00453-024-01287-4\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, SOFTWARE ENGINEERING\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algorithmica","FirstCategoryId":"94","ListUrlMain":"https://link.springer.com/article/10.1007/s00453-024-01287-4","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
We consider the popular matching problem in a roommates instance G on n vertices, i.e., G is a graph where each vertex has a strict preference order over its neighbors. A matching M is popular if there is no matching N such that the vertices that prefer N to M outnumber those that prefer M to N. It is known that it is NP-hard to decide if G admits a popular matching or not. There is no better algorithm known for this problem than the brute force algorithm that enumerates all matchings and tests each for popularity—this could take n! time. Here we show an \(O^*(k^n)\) time algorithm for this problem, where \(k < 7.32\). We use the recent breakthrough result on the maximum number of stable matchings possible in a roommates instance to analyze our algorithm for the popular matching problem. We identify a natural (also, hard) subclass of popular matchings called truly popular matchings that are “popular fractional” and show an \(O^*(2^n)\) time algorithm for the truly popular matching problem in G. We also identify a subclass of max-size popular matchings called super-dominant matchings and show a linear time algorithm for the super-dominant roommates problem.
期刊介绍：
Algorithmica is an international journal which publishes theoretical papers on algorithms that address problems arising in practical areas, and experimental papers of general appeal for practical importance or techniques. The development of algorithms is an integral part of computer science. The increasing complexity and scope of computer applications makes the design of efficient algorithms essential.
Algorithmica covers algorithms in applied areas such as: VLSI, distributed computing, parallel processing, automated design, robotics, graphics, data base design, software tools, as well as algorithms in fundamental areas such as sorting, searching, data structures, computational geometry, and linear programming.
In addition, the journal features two special sections: Application Experience, presenting findings obtained from applications of theoretical results to practical situations, and Problems, offering short papers presenting problems on selected topics of computer science.
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