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The traveling salesman problem might be described as follows: Find the shortest route for a salesman starting from a given city, visiting each of a specified group of cities, and then returning to the original point of departure. More generally, given an n by n symmetric matrix D = (dij), where dij represents the "distance" from i to j, arrange the points in a cyclic order in such a way that the sum of the dij between consecutive points is minimal. Since there are only a finite number of possibilities to consider, the problem is to devise a method of picking out the optimal arrangement which is reasonably efficient for fairly large values of n. Although algorithms have been devised for problems of similar nature, e.g., the optimal assignment problem, little is known about the traveling salesman problem. We do not claim that this note alters the situation very much; what we shall do is outline a way of approaching the problem that sometimes, at least, enables one to find an optimal path and prove it so.
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