{"title":"识别$$\\mathbb{R}^d$$中多距图的复杂性","authors":"G. M. Sokolov","doi":"10.1134/s000143462405016x","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the complexity of recognizing <span>\\(A\\)</span>-distance graphs in <span>\\(\\mathbb{R}^d\\)</span> and prove that for all finite sets <span>\\(A\\)</span> such that any two elements of the set differ by a factor <span>\\(\\ge2\\)</span>, the recognition problem for <span>\\(A\\)</span>-distance graphs is <span>\\(\\mathrm{NP}\\)</span>-hard for any <span>\\(d \\geq 3\\)</span>. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Complexity of Recognizing Multidistance Graphs in $$\\\\mathbb{R}^d$$\",\"authors\":\"G. M. Sokolov\",\"doi\":\"10.1134/s000143462405016x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p> We study the complexity of recognizing <span>\\\\(A\\\\)</span>-distance graphs in <span>\\\\(\\\\mathbb{R}^d\\\\)</span> and prove that for all finite sets <span>\\\\(A\\\\)</span> such that any two elements of the set differ by a factor <span>\\\\(\\\\ge2\\\\)</span>, the recognition problem for <span>\\\\(A\\\\)</span>-distance graphs is <span>\\\\(\\\\mathrm{NP}\\\\)</span>-hard for any <span>\\\\(d \\\\geq 3\\\\)</span>. </p>\",\"PeriodicalId\":18294,\"journal\":{\"name\":\"Mathematical Notes\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-07-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Notes\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1134/s000143462405016x\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Notes","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s000143462405016x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Complexity of Recognizing Multidistance Graphs in $$\mathbb{R}^d$$
Abstract
We study the complexity of recognizing \(A\)-distance graphs in \(\mathbb{R}^d\) and prove that for all finite sets \(A\) such that any two elements of the set differ by a factor \(\ge2\), the recognition problem for \(A\)-distance graphs is \(\mathrm{NP}\)-hard for any \(d \geq 3\).
期刊介绍：
Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.
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