Abstract
In the Simulated Annealing algorithm applied to the Traveling Salesman Problem, the total tour length decreases with temperature. Empirical observation shows that the tours become more structured as the temperature decreases. We quantify this fact by proposing the use of the Shannon information content of the probability distribution function of inter–city step lengths. We find that information increases as the Simulated Annealing temperature decreases. We also propose a practical use of this insight to improve the standard algorithm by switching, at the end of the algorithm, the cost function from the total length to information content. In this way, the final tour should not only be shorter, but also smoother.
Highlights
The Traveling Salesman Problem (TSP) seeks to find the tour order in which to visit all the cities of a given set so that the total length traveled is the smallest possible
In the Simulated Annealing algorithm applied to the Traveling Salesman Problem, the total tour length decreases with temperature
We propose a practical use of this insight to improve the standard algorithm by switching, at the end of the algorithm, the cost function from the total length to information content
Summary
The Traveling Salesman Problem (TSP) seeks to find the tour order in which to visit all the cities of a given set so that the total length traveled is the smallest possible. Perhaps not fully theoretically appealing, this practical approach to the TSP is the one taken by applied mathematicians when providing tours to delivery trucks that need to visit a prescribed set of addresses, or microscope probes that need to reach specific sample locations Practical solutions of these types reduce waste and are of great value to the commercial and/or scientific enterprise. If one picks a tour by a random procedure, first most likely that tour will be long because long tours are common while the short ones are rare, and second the longer tours by its own nature of passing through more between city points are less structured, even less visually pleasing due to presence of a large number of sharp turns, than shorter tours This qualitative description can be made quantitative by appealing to the Negentropy, or information content [6].
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