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The Traveling Salesman and Related Stochastic Problems

In the traveling salesman problem, one must find the length of the shortest closed tour visiting given ``cities''. We study the stochastic version of the problem, taking the locations of cities and the distances separating them to be random variables drawn from an ensemble. We consider first the ensemble where cities are placed in Euclidean space. We investigate how the optimum tour length scales with number of cities and with number of spatial dimensions. We then examine the analytical theory behind the random link ensemble, where distances between cities are independent random variables. Finally, we look at the related geometric issue of nearest neighbor distances, and find some remarkable universalities.