The frequency $K_i$s ($i\in[4,n]$) are studied for symmetric traveling salesman problem ($TSP$) to illustrate the structure properties of the edges in optimal Hamiltonian cycle ($OHC$). A frequency $K_i$ is computed with the set of ${{i}\choose{2}}$ optimal $i$-vertex paths with given endpoints (optimal $i$-vertex paths) in one corresponding $K_i$ in $K_n$. Given an $OHC$ edge related to $K_i$, it has certain frequency bigger than $\frac{1}{2}{{i}\choose{2}}$ in the frequency $K_i$, and that of an ordinary edge not in $OHC$ is smaller than $2(n-3)$. Moreover, given a frequency $K_i$ containing an $OHC$ edge related to $K_n$, the frequency of the $OHC$ edge is bigger than $\frac{1}{2}{{i}\choose{2}}$ in the average case. It also found that the probability that an $OHC$ edge is contained in the optimal $i$-vertex paths increases according to $i\in [4, n]$ or keeps stable if it decreases from $i$ to $i+1\leq n$. As the frequency $K_i$s are used to compute the frequency of an edge, each $OHC$ edge reaches its own peak frequency at $i=P_0$ where $P_0=\frac{n}{2} + 2$ for even $n$ or $\frac{n+1}{2} + 1$ for odd $n$. For each ordinary edge out of $OHC$, the probability that they are contained in the optimal $i$-vertex paths decreases according to $i$, respectively, in the average case. Moreover, the average frequency of an ordinary edge will be smaller than $\frac{1}{2}{{i}\choose{2}}$ if $i \geq 2i_d$ where $i_d$ is the smallest number meeting the condition $\frac{(n-2)(n-3) - (i_d-2)(i_d-3)}{(n-2)(n-3) - (i_d-1)(i_d-2)} \geq \sqrt{1 + \frac{2}{i_d(i_d+1)}}$ and $i_d = O(n^{\frac{4}{7}})$. Based on these findings, an algorithm is presented to find $OHC$ in $O(n^2i_d^42^{2i_d})$ time using dynamic programming. The experiments are executed to verify these findings with the benchmark $TSP$ instances.

翻译：为阐明最优哈密顿环（$OHC$）中边的结构性质，本文针对对称旅行商问题（$TSP$）研究了频率 $K_i$（$i\in[4,n]$）。频率 $K_i$ 通过完全图 $K_n$ 中一个对应 $K_i$ 子图内所有给定端点的最优 $i$ 顶点路径（即最优 $i$ 顶点路径）集合 ${{i}\choose{2}}$ 计算得出。对于与 $K_i$ 相关的 $OHC$ 边，其在频率 $K_i$ 中的特定频率大于 $\frac{1}{2}{{i}\choose{2}}$，而非 $OHC$ 的普通边频率则小于 $2(n-3)$。此外，对于包含与 $K_n$ 相关的 $OHC$ 边的频率 $K_i$，该 $OHC$ 边的频率在平均情况下大于 $\frac{1}{2}{{i}\choose{2}}$。研究还发现，$OHC$ 边被包含在最优 $i$ 顶点路径中的概率随 $i\in [4, n]$ 递增，或在从 $i$ 减小至 $i+1\leq n$ 时保持稳定。由于频率 $K_i$ 用于计算边的频率，每个 $OHC$ 边在 $i=P_0$ 处达到其峰值频率，其中 $P_0=\frac{n}{2} + 2$（$n$ 为偶数时）或 $\frac{n+1}{2} + 1$（$n$ 为奇数时）。对于 $OHC$ 之外的每条普通边，在平均情况下，其被包含在最优 $i$ 顶点路径中的概率随 $i$ 递增而递减。此外，当 $i \geq 2i_d$ 时，普通边的平均频率将小于 $\frac{1}{2}{{i}\choose{2}}$，其中 $i_d$ 是满足条件 $\frac{(n-2)(n-3) - (i_d-2)(i_d-3)}{(n-2)(n-3) - (i_d-1)(i_d-2)} \geq \sqrt{1 + \frac{2}{i_d(i_d+1)}}$ 的最小整数，且 $i_d = O(n^{\frac{4}{7}})$。基于这些发现，本文提出一种利用动态规划在 $O(n^2i_d^42^{2i_d})$ 时间内求解 $OHC$ 的算法，并通过基准 $TSP$ 实例进行了实验验证。