Frequency bounds for edges and paths in optimal Hamiltonian cycle based on frequency Kis

Abstract

Traveling salesman problem ($TSP$) is extensively studied in operations research and computer science. In general, the distances of edges are not helpful for finding edges and paths in optimal Hamiltonian cycle ($OHC$). The frequency bounds for edges and paths in $OHC$ are studied based on frequency $K_i$s ($4\le i\le n$) in $K_n$. A frequency $K_i$ is computed with the optimal $i$-vertex paths with given endpoints (optimal $i$-vertex path for short) in one corresponding $K_i$ in $K_n$. In this paper, the lower frequency bounds for $OHC$ edges and paths in $K_n$ are improved under the constrains of optimal $i$-vertex paths in frequency $K_i$ where $i\ge 5$. As the frequency of an edge is computed with frequency $K_5$s under the constraints of optimal 5-vertex paths, the lower frequency bound for $OHC$ edges is improved from 5 to $\frac{131}{20}$ for small $TSP$. For 3-vertex, 4-vertex, and 5-vertex paths in $OHC$, the lower frequency bounds are derived as $\frac{187}{15}$, $\frac{187}{10}$, and $\frac{131}{5}$, respectively. For a $k$-edge path in $OHC$ if $k\ge 5d$ where $d$ is a small number, the lower frequency bound is $7k$. For big $TSP$, the lower frequency bound for a $k$-edge path in $OHC$ is $7k$ where $k\in [1,n]$. In addition, the average frequency of all $OHC$ edges is bigger than 7 for small $TSP$, and 7.5 for big $TSP$. If the frequency of an edge is computed with frequency $K_i$s containing more vertices, the frequency bounds for $OHC$ edges and paths are approximated. As these frequency bounds are taken as threshold to eliminate ordinary edges and paths excluding from $OHC$, the number of preserved solutions is not bigger than 1 unless $e(i-1)!\le 2(n-1)$ exists where $e$ is the base of natural logarithm. The experiments are implemented with $TSP$ instances to verify the findings.