Thermodynamical approach to the travelling salesman problem: An efficient simulation algorithm. (English) Zbl 0534.90091

Černý, V.

J. Optimization Theory Appl. 45, 41-51 (1985).

We present a Monte Carlo algorithm to find approximate solutions of the travelling salesman problem. The algorithm generates randomly the permutations of the stations of the travelling 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 cat 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.

MSC:

90C35	Programming involving graphs or networks
65C05	Monte Carlo methods
80A10	Classical and relativistic thermodynamics
90C10	Integer programming

Keywords:

importance sampling; Monte Carlo algorithm; approximate solutions; travelling salesman; Boltzmann-Gibbs distribution; analogy with thermodynamics

References:

[1]	Kittel, C.,Thermal Physics, John Wiley and Sons, New York, New York, 1969.
[2]	Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., andTeller, E.,Equation of State Calculations by Fast Computing Machines, Journal of Chemical Physics, Vol. 21, pp. 1087–1092, 1953.