A Binomial Distribution Model for the Traveling Salesman Problem Based on Frequency Quadrilaterals

Abstract

We study the symmetric traveling salesman problem via frequency graphs. One computes the frequency of edges by computing how many times an edge occurs in an optimal path involving four vertices. The edges that are in the Optimal Hamiltonian Cycle ($OHC$) have a higher frequency than most edges that are not in the $OHC$ and thus edges with a low frequency can safely be ignored when searching for the optimal solution. A binomial distribution model is introduced for the symmetric traveling salesman problem based on frequency quadrilaterals. When the frequency of each edge is computed with $N$ frequency quadrilaterals, our model suggests that the minimum frequency of an edge in the $OHC$ is $F_\mathrm{min} =(\epsilon_\mathrm{min}+1)N$ where $\frac{4}{3}+\frac{4}{3(n-2)} < \epsilon_\mathrm{min} < 4$. This suggests a heuristic to reduce the number of edges that need to be considered in the search for the $OHC$ which is to keep only those edges whose frequencies are $\geq F_\mathrm{min}$. We explore this heuristic in several real-world examples. An erratum for this paper has been published on August 2017 (see link on the right)