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 in optimal Hamiltonian cycle ($OHC$). For a given $K_i$ in the complete graph $K_n$, 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 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 probability of an ordinary edge definitely decreases if $i \geq i_d$ where $i_d = O(n^{\frac{4}{7}})$ is the smallest number meeting the inequality $\frac{(n-2)(n-3) - (i_d-2)(i_d-3)}{(n-2)(n-3) - (i_d-1)(i_d-2)} \geq \sqrt{1 + \frac{2}{i_d(i_d+1)}}$. Based on these findings, an algorithm is presented to find $OHC$ in $O(n^2i_d^42^{i_d})$ time using dynamic programming. The experiments are executed to verify these findings with various $TSP$ instances.

翻译：针对对称旅行商问题（$TSP$），研究了频率 $K_i$（$i\in[4,n]$）以刻画最优哈密顿回路（$OHC$）中边的结构性质。在完全图 $K_n$ 的给定子图 $K_i$ 中，频率 $K_i$ 通过该子图中所有 ${{i}\choose{2}}$ 条具有固定端点的最优 $i$ 顶点路径（最优 $i$ 顶点路径）进行计算。当某条边属于 $OHC$ 且关联于 $K_i$ 时，其在频率 $K_i$ 中的频率值恒大于 $\frac{1}{2}{{i}\choose{2}}$；而不属于 $OHC$ 的普通边频率则小于 $2(n-3)$。进一步，若频率 $K_i$ 包含 $K_n$ 中某条 $OHC$ 边，则该 $OHC$ 边的平均频率大于 $\frac{1}{2}{{i}\choose{2}}$。研究还发现，当 $i\in [4, n]$ 递增时，$OHC$ 边被最优 $i$ 顶点路径包含的概率随之增加；若从 $i$ 递减至 $i+1\leq n$，该概率保持稳定。当利用频率 $K_i$ 计算边的频率时，每条 $OHC$ 边在 $i=P_0$ 处达到自身频率峰值，其中 $P_0=\frac{n}{2} + 2$（$n$ 为偶数）或 $\frac{n+1}{2} + 1$（$n$ 为奇数）。对于不在 $OHC$ 中的普通边，其被最优 $i$ 顶点路径包含的平均概率随 $i$ 增大而递减。此外，当 $i \geq i_d$ 时普通边的概率必然下降，其中 $i_d = O(n^{\frac{4}{7}})$ 是满足不等式 $\frac{(n-2)(n-3) - (i_d-2)(i_d-3)}{(n-2)(n-3) - (i_d-1)(i_d-2)} \geq \sqrt{1 + \frac{2}{i_d(i_d+1)}}$ 的最小整数。基于上述发现，提出一种采用动态规划、时间复杂度为 $O(n^2i_d^42^{i_d})$ 的算法来求解 $OHC$。通过多种 $TSP$ 实例的实验验证了上述结论。