AI 中文总结
该研究针对对称旅行商问题，通过分析频率$K_i$的分布特征，提出了一种识别最优哈密顿环外普通边的方法，明确了OHC边与普通边的频率差异规律。
AI 中文摘要
针对对称旅行商问题（TSP），研究了频率$K_i$（$i\in[4,n]$）以刻画最优哈密顿环（OHC）内外边的结构性质。在完全图$K_n$中给定一个$K_i$（$i\in[4,n]$），频率$K_i$通过该$K_i$中所有$\binom{i}{2}$条固定端点的最优$i$顶点路径（最优$i$顶点路径）计算得到。对于$K_i$中的一条OHC边，其在频率$K_i$中的数值大于$\frac{1}{2}\binom{i}{2}$；而OHC外的一条普通边，其在频率$K_i$中的数值小于$\frac{1}{2}\binom{i}{2}$。当边的频率通过频率$K_i$计算时，$K_n$的一条OHC边的平均频率大于$\frac{1}{2}\binom{i}{2}$，这表明$K_n$的OHC边也是包含它的$K_i$的OHC边。研究还发现，基于频率$K_i$，OHC边的频率大于$\frac{1}{2}\binom{i}{2}$的概率随$i\in[4,n]$增大而提升；对于OHC外的普通边，其频率小于$\frac{1}{2}\binom{i}{2}$的概率也随$i$增大而提升。基于这些发现，本文提出了一种用于识别TSP普通边的方法。
英文摘要
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 $TSP$.
Comments29 pages, 4 figures
