The N-City Travelling Salesman Problem: Statistical Mechanics and the Metropolis Algorithm

In any N-city travelling salesman problem there are (N - 1)! / ! 2 . - 2 possible tours. We use the Metropolis algorithm to generate a sequence of such tours. This sequence may be viewed as the random evolution of a physical system in contact with a heat-bath. As the temperature is lowered, the tours generated approach the optimal tour. It appears that for large N one arrives within a few percent of the optimal solution in better than quadratic time.