Probabilistic analysis of the Traveling Salesman Problem to appear as a chapter in a forthcoming book on the TSP

In this chapter we study the Hamiltonian cycle and Traveling Salesman problem from a probabilistic point of view. Here we try to elucidate the properties of typical rather than worst-case examples. Structurally, one hopes to bring out the surprising properties of typical instances. Algorithmically the hope is that one can in some way explain the successful solution of large problems, much larger than that predicted by worst-case analysis. This of course raises the question of what do we mean by typical? The mathematical view of this is to de ne a probability space of instances and study the expected properties of ! drawn from with a given probability measure. Our discussion falls naturally into two parts: the independent case and the Euclidean case. The independent case will include a discussion of the existence of Hamiltonian cycles in various classes of random graphs and digraphs. We will then discuss algorithms for nding Hamiltonian cycles which are both fast and likely to succeed. We include a discussion of extension-rotation constructions and the variance reduction technique of Robinson and Wormald. Following this we consider Traveling Salesman Problems where the coeÆcients are drawn independently. We describe both exact and approximate algorithms. We include a section on open problems. After this we survey stochastic results for the total edge length of the Euclidean TSP. Here the cities are assumed to be points in R d and the distance between points is the usual Euclidean distance. We describe ways to prove a.s. limit theorems and concentration inequalities for the total edge length of the shortest tour on a random sample of size n, as n!1. The focus is on presenting probabilistic techniques which not only describe the behavior of the random Euclidean TSP, but which also describe (or have the potential to describe) the behavior of heuristics. The approach centers around the boundary functional method as well as the isoperimetric methods of Rhee and Talagrand.

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