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Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm

Abstract

We present a Monte Carlo algorithm to find approximate solutions of the traveling salesman problem. The algorithm generates randomly the permutations of the stations of the traveling salesman trip, with probability depending on the length of the corresponding route. Reasoning by analogy with statistical thermodynamics, we use the probability given by the Boltzmann-Gibbs distribution. Surprisingly enough, using this simple algorithm, one can get very close to the optimal solution of the problem or even find the true optimum. We demonstrate this on several examples.

We conjecture that the analogy with thermodynamics can offer a new insight into optimization problems and can suggest efficient algorithms for solving them.

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Communicated by S. E. Dreyfus

The author acknowledges stimulating discussions with J. Pišút concerning the main ideas of the present paper. The author is also indebted to P. Brunovský, J. Černý, M. Hamala, Š. Peško, Š. Znám, and R. Zajac for useful comments.

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Černý, V. Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. J Optim Theory Appl 45, 41–51 (1985). https://doi.org/10.1007/BF00940812

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  • DOI: https://doi.org/10.1007/BF00940812

Key Words

  • Traveling salesman problem
  • Monte Carlo optimization
  • importance sampling