Characterization of the probabilistic traveling salesman problem

We show that stochastic annealing can be successfully applied to gain new results on the probabilistic traveling salesman problem. The probabilistic "traveling salesman" must decide on an a priori order in which to visit n cities (randomly distributed over a unit square) before learning that some cities can be omitted. We find the optimized average length of the pruned tour follows E((L) over bar (pruned))=rootnp(0.872-0.105p)f(np), where p is the probability of a city needing to be visited, and f(np)-->1 as np-->infinity. The average length of the a priori tour (before omitting any cities) is found to follow E(L-a priori)=rootn/pbeta(p), where beta(p)=1/[1.25-0.82 ln(p)] is measured for 0.05less than or equal topless than or equal to0.6. Scaling arguments and indirect measurements suggest that beta(p) tends towards a constant for p