The $k$-Opt and Lin-Kernighan algorithm are two of the most important local search approaches for the Metric TSP. Both start with an arbitrary tour and make local improvements in each step to get a shorter tour. We show that, for any fixed $k\geq 3$, the approximation ratio of the $k$-Opt algorithm for Metric TSP is $O(\sqrt[k]{n})$. Assuming the Erd\H{o}s girth conjecture, we prove a matching lower bound of $\Omega(\sqrt[k]{n})$. Unconditionally, we obtain matching bounds for $k=3,4,6$ and a lower bound of $\Omega(n^{\frac{2}{3k-3}})$. Our most general bounds depend on the values of a function from extremal graph theory and are tight up to a factor logarithmic in the number of vertices unconditionally. Moreover, all the upper bounds also apply to a parameterized version of the Lin-Kernighan algorithm with appropriate parameter. In the end, we show that the approximation ratio of $k$-Opt for Graph TSP is $\Omega\left(\frac{\log(n)}{\log\log(n)}\right)$ and $O\left(\left(\frac{\log(n)}{\log\log(n)}\right)^{\log_2(9)+\epsilon}\right)$ for all $\epsilon>0$.

中文翻译：

---

关于度量和图 TSP 的 $k$-Opt 和 Lin-Kernighan 算法的逼近比

$k$-Opt 和 Lin-Kernighan 算法是 Metric TSP 的两种最重要的局部搜索方法。两者都从任意游览开始，并在每个步骤中进行局部改进以获得更短的游览。我们表明，对于任何固定的 $k\geq 3$，度量 TSP 的 $k$-Opt 算法的近似比为 $O(\sqrt[k]{n})$。假设 Erd\H{o} 的周长猜想，我们证明了 $\Omega(\sqrt[k]{n})$ 的匹配下界。无条件地，我们获得了 $k=3,4,6$ 的匹配边界和 $\Omega(n^{\frac{2}{3k-3}})$ 的下限。我们最一般的边界取决于极值图论中函数的值，并且无条件地严格为顶点数的对数因子。此外，所有上限也适用于具有适当参数的 Lin-Kernighan 算法的参数化版本。到底，