A method to identify the ordinary edges for symmetric traveling salesman problem based on frequency $K_i$s

Abstract

The frequency $K_i$s ($i\in[4,n]$) are studied for symmetric traveling salesman problem ($TSP$) to characterize the structure properties of the edges inside and outside the optimal Hamiltonian cycle ($OHC$). Given a $K_i$ in $K_n$ where $i\in [4,n]$, the frequency $K_i$ is computed with the set of ${{i}\choose{2}}$ optimal $i$-vertex paths with fixed endpoints (optimal $i$-vertex paths) in the $K_i$. Given an $OHC$ edge in a $K_i$, it has a frequency bigger than $\frac{1}{2}{{i}\choose{2}}$ in the frequency $K_i$, and that of an ordinary edge outside the $OHC$ is smaller than $\frac{1}{2}{{i}\choose{2}}$. As the frequency of an edge is computed with the frequency $K_i$s, an $OHC$ edge of $K_n$ has an average frequency bigger than $\frac{1}{2}{{i}\choose{2}}$. It indicates an $OHC$ edge of $K_n$ is also one $OHC$ edge of a $K_i$ containing it. It also found that the probability that an $OHC$ edge has the frequency bigger than $\frac{1}{2}{{i}\choose{2}}$ increases according to $i\in [4, n]$ based on the frequency $K_i$s. For an ordinary edge outside the $OHC$, the probability that it has a frequency smaller than $\frac{1}{2}{{i}\choose{2}}$ increases according to $i$. Based on the findings, a method is given to identify the ordinary edges for $TSP$.