{"title":"Efficient constant-factor approximate enumeration of minimal subsets for monotone properties with weight constraints","authors":"","doi":"10.1016/j.dam.2024.10.014","DOIUrl":null,"url":null,"abstract":"<div><div>A property <span><math><mi>Π</mi></math></span> on a finite set <span><math><mi>U</mi></math></span> is <em>monotone</em> if for every <span><math><mrow><mi>X</mi><mo>⊆</mo><mi>U</mi></mrow></math></span> satisfying <span><math><mi>Π</mi></math></span>, every superset <span><math><mrow><mi>Y</mi><mo>⊆</mo><mi>U</mi></mrow></math></span> of <span><math><mi>X</mi></math></span> also satisfies <span><math><mi>Π</mi></math></span>. Many combinatorial properties can be seen as monotone properties. The problem of finding a subset of <span><math><mi>U</mi></math></span> satisfying <span><math><mi>Π</mi></math></span> with the minimum weight is a central problem in combinatorial optimization. Although many approximate/exact algorithms have been developed to solve this kind of problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building accurate mathematical models on real-world problems. A promising approach to overcome this difficulty is to <em>enumerate</em> multiple small solutions rather than to <em>find</em> a single small solution. To this end, given a weight function <span><math><mrow><mi>w</mi><mo>:</mo><mi>U</mi><mo>→</mo><msub><mrow><mi>Q</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>∈</mo><msub><mrow><mi>Q</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></mrow></math></span>, we devise algorithms that <em>approximately</em> enumerate all minimal subsets of <span><math><mi>U</mi></math></span> with weight at most <span><math><mi>k</mi></math></span> satisfying <span><math><mi>Π</mi></math></span> for various monotone properties <span><math><mi>Π</mi></math></span>, where “approximate enumeration” means that algorithms output all minimal subsets satisfying <span><math><mi>Π</mi></math></span> whose weight is at most <span><math><mi>k</mi></math></span> and may output some minimal subsets satisfying <span><math><mi>Π</mi></math></span> whose weight exceeds <span><math><mi>k</mi></math></span> but is at most <span><math><mrow><mi>c</mi><mi>k</mi></mrow></math></span> for some constant <span><math><mrow><mi>c</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. These algorithms allow us to efficiently enumerate minimal vertex covers, minimal dominating sets in bounded degree graphs, minimal feedback vertex sets, minimal hitting sets in bounded rank hypergraphs, etc., of weight at most <span><math><mi>k</mi></math></span> with constant approximation factors.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X24004451","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
A property on a finite set is monotone if for every satisfying , every superset of also satisfies . Many combinatorial properties can be seen as monotone properties. The problem of finding a subset of satisfying with the minimum weight is a central problem in combinatorial optimization. Although many approximate/exact algorithms have been developed to solve this kind of problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building accurate mathematical models on real-world problems. A promising approach to overcome this difficulty is to enumerate multiple small solutions rather than to find a single small solution. To this end, given a weight function and , we devise algorithms that approximately enumerate all minimal subsets of with weight at most satisfying for various monotone properties , where “approximate enumeration” means that algorithms output all minimal subsets satisfying whose weight is at most and may output some minimal subsets satisfying whose weight exceeds but is at most for some constant . These algorithms allow us to efficiently enumerate minimal vertex covers, minimal dominating sets in bounded degree graphs, minimal feedback vertex sets, minimal hitting sets in bounded rank hypergraphs, etc., of weight at most with constant approximation factors.
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.