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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.

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. The problem was posed centuries ago and has enjoyed constant attention and consistent progress en route for better solutions [1,2]. Generally speaking, the optimal solution can be found for sets containing a few dozens cities. However, for sets larger than that, and due the nonpolynomial growth of tours permutations with set size, solutions to the TSP are approximate: in this case one is satisfied with finding a good tour instead of the best tour. Although 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 thus of great value to the commercial and/or scientific enterprise.

Here we consider the randomly generated TSP instances problem with Euclidean distance [

The last ten years have seen a renewed surge in the interest in the TSP in its original form and variations, and the algorithms developed are so efficient that one doubts anything new can be said on the topic. However, there is an unexploited perspective that can give new insights into the problem and suggest improved algorithms. This perspective is proposed here and it is based on the information content in a given tour. 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 [

The TSP has been optimized by genetic algorithms, direct steepest descendant, insect swarm algorithms, etc. Here we use a program written by us based on the extensively tested Simulated Annealing (SA) adapted to the TSP [

Thus the SA-TSP algorithm generates a tour at each time step. We analyze each such tour and produce histogram of steps—that is, a histogram of all the consecutive-city distances that the traveler takes in the tour.

The mean length is known to be

[

The increased narrowness of the PDFs for shorter tours is a consequence of the smoother nature of better tours. This can be seen in

communicate anything. It seems as if the shortest tours carried more information. Although this is a vague statement, we will make it quantitative in the next section.

In this section we follow Wheeler’s inspiration that “all things physical are information-theoretical in origin” [

where is the probability of having a step length for a given tour of total length

With this definition, we could use the histograms for the tour step lengths introduced in the previous section

however, to gain more insight we shall introduce an analytical description of the problem.

In

where A is a normalization constant, , and.

We then studied the SA TSP from 64 to 2048 cities and found that Equation (3) consistently represents the PDF of tour steps at any stage of the SA annealing algorithm. While the form of Equation (3) remains valid throughout the evolution of the SA, the values of and change as functions of the SA time stage.

Therefore, we take Equation (3) as the PDF of TSP tour steps at any stage of the SA. In addition, the Negentropy of a given PDF cannot depend either on its area under the curve (total tour length) or on its mean step size. The Negentropy, on the other hand, should be dependent on the variance and higher moments since they carry (dis)order information. Thus we consider the area and mean normalized version of Equation (3)

where the parameter, that carries information of the mean, disappeared.

This PDF has already been recognized as that of the near neighbors for a one-dimensional gas with logarithmic pair potential. In addition, it is a particular case (rank 2) of a daisy model, in which the PDF is derived by the Poisson spectrum and then removing every r^{th} intermediate levels, where r defines the rank of the derived distribution [

Using Equation (1) for the PDF in (4) one obtains,

where we have added the arbitrary constant 1 to the definition of Negentropy for later convenience. Then the Negentropy becomes explicitly,

where is the logarithmic derivative function [

In addition to being able to quantify the previously vague concept of tour appearance, can the ideas of Section 3 be of any practical use? We argue here that indeed, we can refine the optimum solution obtained by standard SA TSP.

However, we also see that their convergence rates are different: the total length saturates faster than the negentropy. Therefore, one could run the SA program monitoring the total length until it does not change substantially. At that point one would switch to monitor the negentropy instead. However, while the total length is easy to compute, , the negentropy would require to build a histogram and then use Equation (1). However, this would be too time consuming for the program. Instead, we use the analytical results of Section 3, to expedite the calculations. The variance of the PDF in Equation (4) is

So given the set of step lengths in the program, one computes and then substitutes in

Thus we have a fast way to evaluate information in terms of the firs two moments of the step-length.

The proposed improvement may provide a marginal gain in tour length. But while indeed the total tour length will stay practically unchanged the algorithm will search, out of those many good solutions, for the ones with maximum information. In the end, the traveler will not only travel less, but will also travel more comfortably.

We have introduced a new perspective from which to study the Traveling Salesman Problem. Besides using the total tour length as the cost function, we propose to also use the tour information content. The Shannon entropy has been used here to quantify the intuitive fact that generally, shorter tours contain more information. The analytical expression for the information content provides insight into the TSP structure. We have also proposed a practical improvement to the SA TSP where, when an acceptable short tour has been found by the standard algorithm, the program remains running to search among short tours for those with more information. This provides the additional benefit that the tour selected will not only be short but also smoother. This could be relevant, for example, when the TSP is used to design solutions that require mechanical manipulation of an arm to visit points on a surface. Such could be the case when visiting preset locations on a sample with Atomic Force Microscopy [