Statistical Mechanics algorithm for Monte Carlo optimization

The traveling‐salesman problem is the classic example of a class of recalcitrant combinatorial‐optimization tasks (the “NP‐complete” problems) for whose exact solution the only known algorithms require a number of steps that grows at least exponentially with the number of elements in the problem. Finding, by brute force, the shortest path by which a traveling salesman can complete a tour of N cities requires compiling a list of $(N−1)1/2$ alternative tour lengths—a number that grows faster than any finite power of N and quickly becomes intractable. Finding an exact polynomial algorithm (bounded by a finite power of N) for any one member of the class of NP‐complete problems would constitute a proof that all such problems possess an “efficient” exact algorithm; but it is generally believed that no such algorithm exists.