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An Approximation Algorithm for the Clustered Path Travelling Salesman Problem

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Algorithmic Aspects in Information and Management (AAIM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13513))

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Abstract

In this paper, we consider the clustered path travelling salesman problem. In this problem, we are given a complete graph \(G=(V,E)\) with edge weight satisfying the triangle inequality. In addition, the vertex set V is partitioned into clusters \(V_1,\cdots ,V_k\). The objective of the problem is to find a minimum Hamiltonian path in G, and in the path all vertices of each cluster are visited consecutively. We provide a polynomial-time approximation algorithm for the problem.

Supported by the NSF of China (No. 11971146), the NSF of Hebei Province of China (No. A2019205089, No. A2019205092), Overseas Expertise Introduction Program of Hebei Auspices (25305008) and the Graduate Innovation Grant Program of Hebei Normal University (No. CXZZSS2022052).

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References

  1. An, H.-C., Kleinberg, R., Shmoys, D.B.: Improving Christofide’ algorithm for the s-t path TSP. J. ACM 62(5), 1–28 (2015)

    Article  MathSciNet  Google Scholar 

  2. Anily, S., Bramel, J., Hertz, A.: A 5/3-approximation algorithm for the clustered traveling salesman tour and path problems. Oper. Res. Lett. 24, 29–35 (1999)

    Article  MathSciNet  Google Scholar 

  3. Arkin, E.M., Hassin, R., Klein, L.: Restricted delivery problems on a network. Networks 29, 205–216 (1994)

    Article  MathSciNet  Google Scholar 

  4. Bland, R.G., Shallcross, D.F.: Large travelling salesman problems arising from experiments in X-ray crystallography: a preliminary report on computation. Oper. Res. Lett. 8(3), 125–128 (1989)

    Article  MathSciNet  Google Scholar 

  5. Chisman, J.A.: The clustered traveling salesman problem. Comput. Oper. Res. 2, 115–119 (1975)

    Article  Google Scholar 

  6. Christofides, N.: Worst-case analysis of a new heuristic for the Travelling Salesman Problem. Technical report 388, Graduate School of Industrial Administration, Carnegie Mellon University (1976)

    Google Scholar 

  7. Diestel, R.: Graph Theory. Springer, New York (2017). https://doi.org/10.1007/978-3-662-53622-3

    Book  MATH  Google Scholar 

  8. Edmonds, J.: Paths, trees and flowers. Can. J. Math. 17, 449–467 (1965)

    Article  MathSciNet  Google Scholar 

  9. Gendreau, M., Hertz, A., Laporte, G.: The traveling salesman problem with backhauls. Comput. Oper. Res. 23, 501–508 (1996)

    Article  MathSciNet  Google Scholar 

  10. Grinman, A.: The Hungarian algorithm for weighted bipartite graphs. Seminar in Theoretical Computer Science (2015)

    Google Scholar 

  11. Grötschel, M., Holland, O.: Solution of large-scale symmetric traveling salesman problems. Math. Program. 51, 141–202 (1991)

    Article  Google Scholar 

  12. Gottschalk, C., Vygen, J.: Better s-t-tours by Gao trees. Math. Program. 172, 191–207 (2018)

    Article  MathSciNet  Google Scholar 

  13. Guttmann-Beck, N., Hassin, R., Khuller, S., Raghavachari, B.: Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem. Algorithmica 28, 422–437 (2000)

    Article  MathSciNet  Google Scholar 

  14. Hong, Y.M., Lai, H.J., Liu, Q.H.: Supereulerian digraphs. Discrete Math. 330, 87–95 (2014)

    Article  MathSciNet  Google Scholar 

  15. Hoogeveen, J.A.: Analysis of Christofides’ heuristic: some paths are more difficult than cycles. Oper. Res. Lett. 10, 291–295 (1991)

    Article  MathSciNet  Google Scholar 

  16. Jongens, K., Volgenant, T.: The symmetric clustered traveling salesman problem. Eur. J. Oper. Res. 19, 68–75 (1985)

    Article  MathSciNet  Google Scholar 

  17. Kawasaki, M., Takazawa, T.: Improving approximation ratios for the clustered travelling salesman problem. J. Oper. Res. Soc. Jpn. 63(2), 60–70 (2020)

    MATH  Google Scholar 

  18. Plante, R.D., Lowe, T.J., Chandrasekaran, R.: The product matrix travelling salesman problem: an application and solution heuristics. Oper. Res. 35, 772–783 (1987)

    Article  MathSciNet  Google Scholar 

  19. Sebő, A., van Zuylen, A.: The salesman’s improved paths: a 3/2+1/34 approximation. In: Proceedings of 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 118–127 (2016)

    Google Scholar 

  20. Traub, V., Vygen, J.: Approaching 3/2 for the s-t path TSP. J. ACM 66(2), 1–17 (2019)

    Article  MathSciNet  Google Scholar 

  21. Yao, A.: An O\((\left|E\right|\log \log \left|V\right|)\) algorithm for finding minimum spanning trees. Inf. Process. Lett. 4, 21–23 (1975)

    Article  Google Scholar 

  22. Zenklusen, R.: A 1.5-approximation for path TSP. In: Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1539–1549 (2019)

    Google Scholar 

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Authors and Affiliations

  1. Hebei Key Laboratory of Computational Mathematics and Applications, School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China

    Jiaxuan Zhang, Suogang Gao, Bo Hou & Wen Liu

Authors
  1. Jiaxuan Zhang
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  2. Suogang Gao
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  3. Bo Hou
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  4. Wen Liu
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Corresponding author

Correspondence to Wen Liu .

Editor information

Editors and Affiliations

  1. Guangdong University of Technology, Guangzhou, China

    Qiufen Ni

  2. University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

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Zhang, J., Gao, S., Hou, B., Liu, W. (2022). An Approximation Algorithm for the Clustered Path Travelling Salesman Problem. In: Ni, Q., Wu, W. (eds) Algorithmic Aspects in Information and Management. AAIM 2022. Lecture Notes in Computer Science, vol 13513. Springer, Cham. https://doi.org/10.1007/978-3-031-16081-3_2

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  • DOI: https://doi.org/10.1007/978-3-031-16081-3_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-16080-6

  • Online ISBN: 978-3-031-16081-3

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