Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

An analogue approach to the travelling salesman problem using an elastic net method

Abstract

The travelling salesman problem1 is a classical problem in the field of combinatorial optimization, concerned with efficient methods for maximizing or minimizing a function of many independent variables. Given the positions of N cities, which in the simplest case lie in the plane, what is the shortest closed tour in which each city can be visited once? We describe how a parallel analogue algorithm, derived from a formal model2–3 for the establishment of topographically ordered projections in the brain4–10, can be applied to the travelling salesman problem1,11,12. Using an iterative procedure, a circular closed path is gradually elongated non-uniformly until it eventually passes sufficiently near to all the cities to define a tour. This produces shorter tour lengths than another recent parallel analogue algorithm13, scales well with the size of the problem, and is naturally extendable to a large class of optimization problems involving topographic mappings between geometrical structures14.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Prices may be subject to local taxes which are calculated during checkout

References

  1. Lawler, E. L., Lenstra, J. K., Rinooy Khan, A. H. G. & Shmoys, D. B. (eds) The Traveling Salesman Problem (Wiley, New York, 1985).

  2. Von der Malsburg, Ch. & Willshaw, D. J. Proc. natn. Acad. Sci. U.S.A. 74, 5176–5178 (1977).

    Article  ADS  CAS  Google Scholar 

  3. Willshaw, D. J. & Von der Malsburg, Ch. Phil. Trans. R. Soc. B287, 203–243 (1979).

    Article  CAS  Google Scholar 

  4. Talbot, S. A. & Marshall, W. H. Am. J. Ophthal. 24, 1255–1264 (1941).

    Article  Google Scholar 

  5. Gaze, R. M. The Formation of Nerve Connections (Academic, London, 1970).

    MATH  Google Scholar 

  6. Cowan, W. M. & Hunt, R. K. in The Molecular Basis of Neural Development (eds Edelman, G. M., Gall, W. E. & Cowan, W. M.) 389–428 (Wiley, New York, 1986).

    Google Scholar 

  7. Rose, J. E., Galambos, R. & Hughes, J. R. Bull. Johns Hopkins Hosp. 104, 211–251 (1959).

    CAS  PubMed  Google Scholar 

  8. Knudsen, E. I. & Konishi, M. Science 200, 795–797 (1978).

    Article  ADS  CAS  Google Scholar 

  9. Drager, U. C. & Hubel, D. H. J. Neurophysiol. 38, 690–713 (1975).

    Article  CAS  Google Scholar 

  10. Wurtz, R. H. & Albano, J. E. A. Rev. Neurosci. 3, 189–226 (1980).

    Article  CAS  Google Scholar 

  11. Garey, M. R. & Johnson, D. S. Computers and Intractability (Freeman, San Francisco, 1979).

    MATH  Google Scholar 

  12. Papadimitriou, C. H. Theor. Comput. Sci. 4, 237–244 (1977).

    Article  Google Scholar 

  13. Hopfield, J. J. & Tank, D. W. Biol. Cybern. 5, 141–152 (1985).

    Google Scholar 

  14. Mitchison, G. J. & Durbin, R. SIAM J. alg. disc. Meth. 7, 571–582 (1986).

    Article  Google Scholar 

  15. Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Science 220, 671–680 (1983).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  16. Kirkpatrick, S. J. statist. Phys. 34, 975–986 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  17. Lin, S. & Kernighan, B. W. Oper. Res. 21, 498–516 (1973).

    Article  Google Scholar 

  18. Leuenberger, D. G. Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, Massachusetts, 1984).

    Google Scholar 

  19. Christofides, N. in The Traveling Salesman Problem (eds Lawler, E. L., Lenstra, J. K., Rinnooy Khan, A. H. G. & Shmoys, D. B.) 431–448 (Wiley, New York, 1985).

    Google Scholar 

  20. Julesz, B. Foundations of Cyclopean Vision (University of Chicago, 1971).

    Google Scholar 

  21. Marr, D. & Poggio, T. Proc. R. Soc B204, 301–328 (1979).

    CAS  Google Scholar 

  22. Hubel, D. H. & Wiesel, T. N. Proc. R. Soc. B198, 1–59 (1977).

    ADS  Google Scholar 

  23. Law, M. I. & Constantine-Paton, M. J. Neurosci. 7, 741–759 (1981).

    Article  Google Scholar 

  24. Fawcett, J. W. & Willshaw, D. J. Nature 296, 350–352 (1982).

    Article  ADS  CAS  Google Scholar 

  25. Von der Malsburg, Ch. & Willshaw, D. J. Expl Brain Res. (Suppl.) 1, 463–469 (1976).

    Google Scholar 

  26. Von der Malsburg, Ch. Biol. Cybern. 32, 49–62 (1979).

    Article  CAS  Google Scholar 

  27. Swindale, N. V. Proc. R. Soc. B208, 243–264 (1980).

    ADS  CAS  Google Scholar 

  28. Kohonen, T. Self-organisation and Associative Memory (Springer, Berlin, 1984).

    MATH  Google Scholar 

  29. Brady, R. M. Nature 317, 804–806 (1985).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Durbin, R., Willshaw, D. An analogue approach to the travelling salesman problem using an elastic net method. Nature 326, 689–691 (1987). https://doi.org/10.1038/326689a0

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/326689a0

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing