{"title":"区间图和和弦图上最大独立集最小度贪婪算法的近似率","authors":"","doi":"10.1016/j.dam.2024.09.009","DOIUrl":null,"url":null,"abstract":"<div><p>In this article we prove that the minimum-degree greedy algorithm, with adversarial tie-breaking, is a <span><math><mrow><mo>(</mo><mn>2</mn><mo>/</mo><mn>3</mn><mo>)</mo></mrow></math></span>-approximation for the <span>Maximum Independent Set</span> problem on interval graphs. We show that this is tight, even on unit interval graphs of maximum degree 3. We show that on chordal graphs, the greedy algorithm is a <span><math><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></math></span>-approximation and that this is again tight. These results contrast with the known (tight) approximation ratio of <span><math><mfrac><mrow><mn>3</mn></mrow><mrow><mi>Δ</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math></span> of the greedy algorithm for general graphs of maximum degree <span><math><mi>Δ</mi></math></span>.</p></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0166218X24003986/pdfft?md5=3034910c4aa5dd66c10a2331bea125fc&pid=1-s2.0-S0166218X24003986-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Approximation ratio of the min-degree greedy algorithm for Maximum Independent Set on interval and chordal graphs\",\"authors\":\"\",\"doi\":\"10.1016/j.dam.2024.09.009\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this article we prove that the minimum-degree greedy algorithm, with adversarial tie-breaking, is a <span><math><mrow><mo>(</mo><mn>2</mn><mo>/</mo><mn>3</mn><mo>)</mo></mrow></math></span>-approximation for the <span>Maximum Independent Set</span> problem on interval graphs. We show that this is tight, even on unit interval graphs of maximum degree 3. We show that on chordal graphs, the greedy algorithm is a <span><math><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></math></span>-approximation and that this is again tight. These results contrast with the known (tight) approximation ratio of <span><math><mfrac><mrow><mn>3</mn></mrow><mrow><mi>Δ</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math></span> of the greedy algorithm for general graphs of maximum degree <span><math><mi>Δ</mi></math></span>.</p></div>\",\"PeriodicalId\":50573,\"journal\":{\"name\":\"Discrete Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-09-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0166218X24003986/pdfft?md5=3034910c4aa5dd66c10a2331bea125fc&pid=1-s2.0-S0166218X24003986-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0166218X24003986\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X24003986","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Approximation ratio of the min-degree greedy algorithm for Maximum Independent Set on interval and chordal graphs
In this article we prove that the minimum-degree greedy algorithm, with adversarial tie-breaking, is a -approximation for the Maximum Independent Set problem on interval graphs. We show that this is tight, even on unit interval graphs of maximum degree 3. We show that on chordal graphs, the greedy algorithm is a -approximation and that this is again tight. These results contrast with the known (tight) approximation ratio of of the greedy algorithm for general graphs of maximum degree .
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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