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 启发式算法是旅行商问题的一种简单改进型启发式算法。该算法从任意一条回路开始，反复用 $k$ 条其他边替换回路中的 $k$ 条边，只要替换后回路长度更短。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，还可推广到 $1 \le p < \infty$ 的任意 $p$-范数。