{"title":"Lower bounds on the integrality ratio of the subtour LP for the traveling salesman problem","authors":"Xianghui Zhong","doi":"10.1016/j.dam.2024.12.029","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper we investigate instances with high integrality ratio of the subtour LP. We determine the instances maximizing the integrality ratio for Rectilinear TSP with up to 10 vertices and for Multidimensional Rectilinear TSP with up to 12 vertices. Based on these instances we give families of instances whose integrality ratio converges to <span><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> for <span>Rectilinear</span>, <span>Multidimensional Rectilinear</span> and <span>Euclidean TSP</span> that have similar structures.</div><div>We also investigate the concept of local optimality with respect to integrality ratio and develop several algorithms to find instances with high integrality ratio. Furthermore, we describe a family of instances that are hard to solve in practice. The currently fastest TSP solver <span>Concorde</span> needs more than two days to solve an instance from the family with 52 vertices.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 109-129"},"PeriodicalIF":1.0000,"publicationDate":"2025-04-15","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/S0166218X2400547X","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/1/13 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we investigate instances with high integrality ratio of the subtour LP. We determine the instances maximizing the integrality ratio for Rectilinear TSP with up to 10 vertices and for Multidimensional Rectilinear TSP with up to 12 vertices. Based on these instances we give families of instances whose integrality ratio converges to for Rectilinear, Multidimensional Rectilinear and Euclidean TSP that have similar structures.
We also investigate the concept of local optimality with respect to integrality ratio and develop several algorithms to find instances with high integrality ratio. Furthermore, we describe a family of instances that are hard to solve in practice. The currently fastest TSP solver Concorde needs more than two days to solve an instance from the family with 52 vertices.
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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