The $k$-Opt heuristic is a simple improvement heuristic for the Traveling Salesman Problem. It starts with an arbitrary tour and then repeatedly replaces $k$ edges of the tour by $k$ other edges, as long as this yields a shorter tour. We will prove that for 2-dimensional Euclidean Traveling Salesman Problems with $n$ cities the approximation ratio of the $k$-Opt heuristic is $\Theta(\log n / \log \log n)$. This improves the upper bound of $O(\log n)$ given by Chandra, Karloff, and Tovey in 1999 and provides for the first time a non-trivial lower bound for the case $k\ge 3$. Our results not only hold for the Euclidean norm but extend to arbitrary $p$-norms with $1 \le p < \infty$.

翻译：$k$-Opt启发式算法是一种用于旅行商问题的简单改进型启发式算法。该算法从任意一条环路出发，不断用$k$条其他边替换当前环路中的$k$条边，只要这样能获得更短的环路。我们将证明：对于包含$n$个城市的二维欧几里得旅行商问题，$k$-Opt启发式算法的近似比为$\Theta(\log n / \log \log n)$。这一结果改进了Chandra、Karloff和Tovey于1999年给出的$O(\log n)$上界，并首次为$k\ge 3$的情形提供了非平凡下界。我们的结论不仅适用于欧几里得范数，还可推广至满足$1 \le p < \infty$的任意$p$-范数。