{"title":"用有界循环数覆盖图的非对称问题的多项式时间逼近性","authors":"M. Yu. Khachai, E. D. Neznakhina, K. V. Ryzhenko","doi":"10.1134/s008154382306010x","DOIUrl":null,"url":null,"abstract":"<p>Recently, O. Svensson and V. Traub have provided the first proof of the polynomial-time approximability of the asymmetric traveling salesman problem (ATSP) in the class of constant-factor approximation algorithms. Just as the famous Christofides–Serdyukov algorithm for the symmetric routing problems, these breakthrough results, applied as a “black box,” have opened an opportunity for developing the first constant-factor polynomial-time approximation algorithms for several related combinatorial problems. At the same time, problems have been revealed in which this simple approach, based on reducing a given instance to one or more auxiliary ATSP instances, does not succeed. In the present paper, we extend the Svensson–Traub approach to the wider class of problems related to finding a minimum-weight cycle cover of an edge-weighted directed graph with an additional constraint on the number of cycles. In particular, it is shown for the first time that the minimum weight cycle cover problem with at most <span>\\(k\\)</span> cycles admits polynomial-time approximation with constant factor <span>\\(\\max\\{22+\\varepsilon,4+k\\}\\)</span> for arbitrary <span>\\(\\varepsilon>0\\)</span>.\n</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Polynomial-Time Approximability of the Asymmetric Problem of Covering a Graph by a Bounded Number of Cycles\",\"authors\":\"M. Yu. Khachai, E. D. Neznakhina, K. V. Ryzhenko\",\"doi\":\"10.1134/s008154382306010x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Recently, O. Svensson and V. Traub have provided the first proof of the polynomial-time approximability of the asymmetric traveling salesman problem (ATSP) in the class of constant-factor approximation algorithms. Just as the famous Christofides–Serdyukov algorithm for the symmetric routing problems, these breakthrough results, applied as a “black box,” have opened an opportunity for developing the first constant-factor polynomial-time approximation algorithms for several related combinatorial problems. At the same time, problems have been revealed in which this simple approach, based on reducing a given instance to one or more auxiliary ATSP instances, does not succeed. In the present paper, we extend the Svensson–Traub approach to the wider class of problems related to finding a minimum-weight cycle cover of an edge-weighted directed graph with an additional constraint on the number of cycles. In particular, it is shown for the first time that the minimum weight cycle cover problem with at most <span>\\\\(k\\\\)</span> cycles admits polynomial-time approximation with constant factor <span>\\\\(\\\\max\\\\{22+\\\\varepsilon,4+k\\\\}\\\\)</span> for arbitrary <span>\\\\(\\\\varepsilon>0\\\\)</span>.\\n</p>\",\"PeriodicalId\":54557,\"journal\":{\"name\":\"Proceedings of the Steklov Institute of Mathematics\",\"volume\":\"22 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-02-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Steklov Institute of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1134/s008154382306010x\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Steklov Institute of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s008154382306010x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Polynomial-Time Approximability of the Asymmetric Problem of Covering a Graph by a Bounded Number of Cycles
Recently, O. Svensson and V. Traub have provided the first proof of the polynomial-time approximability of the asymmetric traveling salesman problem (ATSP) in the class of constant-factor approximation algorithms. Just as the famous Christofides–Serdyukov algorithm for the symmetric routing problems, these breakthrough results, applied as a “black box,” have opened an opportunity for developing the first constant-factor polynomial-time approximation algorithms for several related combinatorial problems. At the same time, problems have been revealed in which this simple approach, based on reducing a given instance to one or more auxiliary ATSP instances, does not succeed. In the present paper, we extend the Svensson–Traub approach to the wider class of problems related to finding a minimum-weight cycle cover of an edge-weighted directed graph with an additional constraint on the number of cycles. In particular, it is shown for the first time that the minimum weight cycle cover problem with at most \(k\) cycles admits polynomial-time approximation with constant factor \(\max\{22+\varepsilon,4+k\}\) for arbitrary \(\varepsilon>0\).
期刊介绍:
Proceedings of the Steklov Institute of Mathematics is a cover-to-cover translation of the Trudy Matematicheskogo Instituta imeni V.A. Steklova of the Russian Academy of Sciences. Each issue ordinarily contains either one book-length article or a collection of articles pertaining to the same topic.