Optimizing with Attractor Combinatorial Complexity of the Traveling Salesman Problem

Optimizing with Attractor Combinatorial Complexity of the Traveling Salesman Problem | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Optimizing with Attractor Combinatorial Complexity of the Traveling Salesman Problem Weiqi Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3822767/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper studies combinatorial complexity of the Traveling Salesman Problem (TSP) from the attractor perspective. In order to find the exact optimal tour for a TSP instance, a kind of exhaustive search is unavoidable. However, the combinatorial explosion of possible tours in the solution space makes an exhaustive search algorithm infeasible for large instances. How can we reduce the search space efficiently and effectively to make an exhaustive search algorithm feasible? This is the question this paper aims to answer. The TSP has a data structure, through which its complexity can be exponentially reduced by a polynomial-time algorithm. This paper describes the complexity reducibility of the TSP from the attractor perspective of dynamical systems. A local search algorithm can be used to reduce the search space from the entire solution space to a much smaller attractor for an exhaustive search. The results show that the TSP might not be as complex as we have expected. Mathematics Subject Classification (2020) 05A15 · 05C05 · 05C30 ·37A50 · 37N40 · 68Q17 · 68Q25 · 68W40 · 90C27 Theoretical Computer Science Combinatorial optimization Computational complexity Traveling salesman problem Analysis of algorithm Dynamical systems Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3822767","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":264418344,"identity":"8aecdb73-b03c-471d-9a8f-25043c1867b0","order_by":0,"name":"Weiqi Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAp0lEQVRIiWNgGAWjYFAC5oYDHxgYeBgkGEAkUYCx4eAMkrUwg1USrcXgeGPjYdu2bTL8sxsYH7xtI0bLmYMNh3PbbvNI3DnAbDiXGC1mNxIhWgwkEtikeYnScv9hw2FLiBb238RpucHYcJgRagszUVrszyQ2HOw5B/TLjcRmyTnniNAi2X748IcfZbft+WckH/zwpowILUiAsYE09aNgFIyCUTAKcAMAVww5X9+wivoAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0002-0739-6859","institution":"University of Michigan-Flint","correspondingAuthor":true,"prefix":"","firstName":"Weiqi","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2023-12-30 00:57:57","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-3822767/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3822767/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":49097441,"identity":"aeeed569-735f-4ed1-9dce-12b2228a4845","added_by":"auto","created_at":"2024-01-03 04:18:32","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":999379,"visible":true,"origin":"","legend":"","description":"","filename":"JCOPaper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3822767/v1_covered_db6b579e-1534-44d3-bcf9-f0dd051187c5.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eOptimizing with Attractor Combinatorial Complexity of the Traveling Salesman Problem\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"University of Michigan–Flint","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Combinatorial optimization, Computational complexity, Traveling salesman problem, Analysis of algorithm, Dynamical systems","lastPublishedDoi":"10.21203/rs.3.rs-3822767/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3822767/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper studies combinatorial complexity of the Traveling Salesman Problem (TSP) from the attractor perspective. In order to find the exact optimal tour for a TSP instance, a kind of exhaustive search is unavoidable. However, the combinatorial explosion of possible tours in the solution space makes an exhaustive search algorithm \u003cu\u003einfeasible\u003c/u\u003e for large instances. How can we reduce the search space efficiently and effectively to make an exhaustive search algorithm feasible? This is the question this paper aims to answer. The TSP has a data structure, through which its complexity can be exponentially reduced by a polynomial-time algorithm. This paper describes the complexity \u003cu\u003ereducibility\u003c/u\u003e of the TSP from the attractor perspective of dynamical systems. A local search algorithm can be used to reduce the search space from the entire solution space to a much smaller attractor for an exhaustive search. 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