2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on real world Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every $p\in\mathbb{N}$, a family of $L_p$ instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which $n$ points are placed uniformly at random in the unit square $[0,1]^2$. We consider a more advanced model in which the points can be placed independently according to general distributions on $[0,1]^d$, for an arbitrary $d\ge 2$. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number $n$ of points and the maximal density $\phi$ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of $\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})$. When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to $\tilde{O}(n^{4+1/3-1/d}\cdot\phi^{8/3})$. If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by $\tilde{O}(n^{4-1/d}\cdot\phi)$. In addition, we prove an upper bound of $O(\sqrt[d]{\phi})$ on the expected approximation factor with respect to all $L_p$ metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with $\phi=1$ and a smoothed analysis with Gaussian perturbations of standard deviation $\sigma$ with $\phi\sim1/\sigma^d$.

翻译：2-Opt可能是旅行商问题中最基本的局部搜索启发式算法。该启发式算法在真实世界的欧几里得实例上，无论是在运行时间还是近似比方面，都取得了惊人优异的结果。关于2-Opt性能的实验研究众多，然而对该启发式算法的理论认识仍然非常有限。甚至连其在二维欧几里得实例上的最坏情况运行时间此前也未知。本文通过以下方式阐明该问题：针对每个 $p\in\mathbb{N}$，提出一族 $L_p$ 实例，在这些实例上2-Opt算法可能采取指数级步骤。先前的概率分析仅限于将 $n$ 个点均匀随机放置在单位正方形 $[0,1]^2$ 中的实例。本文考虑一种更高级的模型，其中点可以根据 $[0,1]^d$ 上的一般分布独立放置，且 $d\ge 2$ 任意。特别地，我们允许不同点服从不同分布。我们研究了关于点数 $n$ 和概率分布最大密度 $\phi$ 的局部改进预期次数。我们证明了任何2-Opt改进路径的预期长度上界为 $\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})$。当以插入启发式算法计算的初始环游作为起点时，预期步数的上界进一步提升至 $\tilde{O}(n^{4+1/3-1/d}\cdot\phi^{8/3})$。若距离按曼哈顿度量测量，则预期步数上界为 $\tilde{O}(n^{4-1/d}\cdot\phi)$。此外，我们证明了关于所有 $L_p$ 度量的预期近似比上界为 $O(\sqrt[d]{\phi})$。需要指出的是，我们的概率分析涵盖了 $\phi=1$ 的均匀输入模型以及标准偏差为 $\sigma$ 且 $\phi\sim1/\sigma^d$ 的高斯扰动平滑分析作为特例。