{"title":"《Diaby’s》等的反例。旅行商问题的线性规划解","authors":"Radosław Hofman","doi":"10.1155/cplx/3672180","DOIUrl":null,"url":null,"abstract":"<div>\n <p>This article presents a method of constructing counter-examples and a complete counter-example to the linear programming model alleged to be the solution to the traveling salesman problem. The counter-example is checked against the model proposed by Diaby et al. However, it applies to all similar formulations of the TSP problem.</p>\n <p>Although the model in question was published in 2006, and there were several discussions regarding its correctness, the counter-example was never presented.</p>\n <p>The presented counter-example is a regular graph, and the aim was not to have an example with the least possible size; therefore, the focus was on clarity. The counter-example has, therefore, 366 nodes in two main clusters, each node (in the main part) having exactly four connections to other nodes in the cluster.</p>\n </div>","PeriodicalId":50653,"journal":{"name":"Complexity","volume":"2025 1","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1155/cplx/3672180","citationCount":"0","resultStr":"{\"title\":\"Counter-Example to Diaby’s et al. Linear Programming Solution to the Traveling Salesman Problem\",\"authors\":\"Radosław Hofman\",\"doi\":\"10.1155/cplx/3672180\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>\\n <p>This article presents a method of constructing counter-examples and a complete counter-example to the linear programming model alleged to be the solution to the traveling salesman problem. The counter-example is checked against the model proposed by Diaby et al. However, it applies to all similar formulations of the TSP problem.</p>\\n <p>Although the model in question was published in 2006, and there were several discussions regarding its correctness, the counter-example was never presented.</p>\\n <p>The presented counter-example is a regular graph, and the aim was not to have an example with the least possible size; therefore, the focus was on clarity. The counter-example has, therefore, 366 nodes in two main clusters, each node (in the main part) having exactly four connections to other nodes in the cluster.</p>\\n </div>\",\"PeriodicalId\":50653,\"journal\":{\"name\":\"Complexity\",\"volume\":\"2025 1\",\"pages\":\"\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2025-01-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1155/cplx/3672180\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Complexity\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1155/cplx/3672180\",\"RegionNum\":4,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Complexity","FirstCategoryId":"5","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1155/cplx/3672180","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Counter-Example to Diaby’s et al. Linear Programming Solution to the Traveling Salesman Problem
This article presents a method of constructing counter-examples and a complete counter-example to the linear programming model alleged to be the solution to the traveling salesman problem. The counter-example is checked against the model proposed by Diaby et al. However, it applies to all similar formulations of the TSP problem.
Although the model in question was published in 2006, and there were several discussions regarding its correctness, the counter-example was never presented.
The presented counter-example is a regular graph, and the aim was not to have an example with the least possible size; therefore, the focus was on clarity. The counter-example has, therefore, 366 nodes in two main clusters, each node (in the main part) having exactly four connections to other nodes in the cluster.
期刊介绍：
Complexity is a cross-disciplinary journal focusing on the rapidly expanding science of complex adaptive systems. The purpose of the journal is to advance the science of complexity. Articles may deal with such methodological themes as chaos, genetic algorithms, cellular automata, neural networks, and evolutionary game theory. Papers treating applications in any area of natural science or human endeavor are welcome, and especially encouraged are papers integrating conceptual themes and applications that cross traditional disciplinary boundaries. Complexity is not meant to serve as a forum for speculation and vague analogies between words like “chaos,” “self-organization,” and “emergence” that are often used in completely different ways in science and in daily life.
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