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 $E\left(L_{\mathrm{pruned}}\right)=\sqrt{\mathrm{np}}(0.872\mathrm{}-0.105p)f\left(\mathrm{np}\right),$ where p is the probability of a city needing to be visited, and $f\left(\mathrm{np}\right)\mathrm{}\to 1$ as $n\overrightarrow{p}\infty .$ The average length of the a priori tour (before omitting any cities) is found to follow ${E(L}_{a\mathrm{priori}})=\sqrt{n/p}\beta \mathrm{}(p),$ where $\beta \mathrm{}\left(p\right)=1/\left[1.25-0.82\ln \mathrm{}\right(p\left)\mathrm{}\right]$ is measured for $0.05<~p<~\mathrm{0.6.}$ Scaling arguments and indirect measurements suggest that $\beta \mathrm{}\left(p\right)\mathrm{}$ tends towards a constant for $p<\mathrm{}\mathrm{0.03.}\mathrm{}$ Our stochastic annealing algorithm is based on limited sampling of the pruned tour lengths, exploiting the sampling error to provide the analog of thermal fluctuations in simulated (thermal) annealing. The method has general application to the optimization of functions whose cost to evaluate rises with the precision required.