Vladimir G. Deineko , Bettina Klinz , Gerhard J. Woeginger
{"title":"The bipartite travelling salesman problem: A pyramidally solvable case","authors":"Vladimir G. Deineko , Bettina Klinz , Gerhard J. Woeginger","doi":"10.1016/j.orl.2024.107130","DOIUrl":null,"url":null,"abstract":"<div><p>Given <em>k</em> blue cities, <em>k</em> red cities and a <span><math><mn>2</mn><mi>k</mi><mo>×</mo><mn>2</mn><mi>k</mi></math></span> distance matrix, the task in the bipartite travelling salesman problem is to find a shortest tour which alternately visits blue and red cities. We consider the special case of Van der Veen distance matrices and show that it remains NP-hard in general but can be solved in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time when all vertices with odd indices are blue and all with even indices are red.</p></div>","PeriodicalId":54682,"journal":{"name":"Operations Research Letters","volume":"55 ","pages":"Article 107130"},"PeriodicalIF":0.8000,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S016763772400066X/pdfft?md5=c5ebab7f8ed20ec28056efd78f98f6f7&pid=1-s2.0-S016763772400066X-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Operations Research Letters","FirstCategoryId":"91","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S016763772400066X","RegionNum":4,"RegionCategory":"管理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"OPERATIONS RESEARCH & MANAGEMENT SCIENCE","Score":null,"Total":0}
引用次数: 0
Abstract
Given k blue cities, k red cities and a distance matrix, the task in the bipartite travelling salesman problem is to find a shortest tour which alternately visits blue and red cities. We consider the special case of Van der Veen distance matrices and show that it remains NP-hard in general but can be solved in time when all vertices with odd indices are blue and all with even indices are red.
期刊介绍:
Operations Research Letters is committed to the rapid review and fast publication of short articles on all aspects of operations research and analytics. Apart from a limitation to eight journal pages, quality, originality, relevance and clarity are the only criteria for selecting the papers to be published. ORL covers the broad field of optimization, stochastic models and game theory. Specific areas of interest include networks, routing, location, queueing, scheduling, inventory, reliability, and financial engineering. We wish to explore interfaces with other fields such as life sciences and health care, artificial intelligence and machine learning, energy distribution, and computational social sciences and humanities. Our traditional strength is in methodology, including theory, modelling, algorithms and computational studies. We also welcome novel applications and concise literature reviews.