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Sufficient and Necessary Conditions for an Edge in the Optimal Hamiltonian Cycle Based on Frequency Quadrilaterals

  • Yong WangEmail author
Article
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Abstract

The symmetric traveling salesman problem is studied according to frequency graphs computed with frequency quadrilaterals. Here, we provide the sufficient and necessary conditions for an optimal Hamiltonian cycle edge based on frequency quadrilaterals. If the probability that an edge has frequency 5 in a frequency quadrilateral is 1, it belongs to the optimal Hamiltonian cycle. For an optimal Hamiltonian cycle edge, the probability that it has frequency 5 in a frequency quadrilateral tends to 1 as the scale of traveling salesman problem is sufficiently large.

Keywords

Traveling salesman problem Optimal Hamiltonian cycle Frequency quadrilateral Sufficient condition Necessary condition 

Mathematics Subject Classification

65K10 68R10 

Notes

Acknowledgements

The authors acknowledge the anonymous referees for their helpful comments to improve the presentation of the paper. We acknowledge Reinelt, G., et al. who provided the TSP data to the TSPLIB, and Cook, W. and Mittelmann, H. who provided the online Concorde. The authors acknowledge the funds supported by the Fundamental Research Funds for the Central Universities (Nos. 2018MS039 and 2018ZD09) and National Natural Science Foundation of China (No. 51205129).

References

  1. 1.
    Gutin, G., Punnen, A.P. (eds.): The Traveling Salesman Problem and Its Variations. Springer, New York (2007)zbMATHGoogle Scholar
  2. 2.
    Karp, R.M.: On the computational complexity of combinatorial problems. Networks(USA) 5, 45–68 (1975)CrossRefzbMATHGoogle Scholar
  3. 3.
    Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. Soc. Ind. Appl. Math. 10, 196–210 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bellman, R.: Dynamic programming treatment of the travelling salesman problem. J. ACM 9, 61–63 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de Klerk, E., Dobre, C.: A comparison of lower bounds for the symmetric circulant traveling salesman problem. Discrete Appl. Math. 159, 1815–1826 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Applegate, D.L., Bixby, R.E., Chvatal, V., Cook, W., Espinoza, D.G., Goycoolea, M., Helsgaun, K.: Certification of an optimal TSP tour through 85900 cities. Oper. Res. Lett. 37, 11–15 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Xu, Z., Rodrigues, B.: An extension of the Christofides heuristic for the generalized multiple depot multiple traveling salesmen problem. Eur. J. Oper. Res. 257, 735–745 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45, 753–782 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lampis, M.: Improved inapproximability for TSP. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) Lecture Notes in Computer Science, vol. 7408, pp. 243–253. Springer, Heidelberg (2012)Google Scholar
  10. 10.
    Wang, Y., Remmel, J.B.: A binomial distribution model for the traveling salesman problem based on frequency quadrilaterals. J. Gr. Algorithms Appl. 20, 411–434 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Wang, Y., Remmel, J.B.: An iterative algorithm to eliminate edges for traveling salesman problem based on a new binomial distribution. Appl. Intell. 48, 4470–4484 (2018)CrossRefGoogle Scholar
  12. 12.
    Hougardy, S., Schroeder, R.T.: Edges elimination in TSP instances. In: Kratsch, D., Todinca, I. (eds.) Lecture Notes in Computer Science, vol. 8747, pp. 275–286. Springer, Heidelberg (2014)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.North China Electric Power UniversityBeijingChina

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