The 2-opt heuristic is a simple local search heuristic for the Travelling Salesperson Problem (TSP). Although it usually performs well in practice, its worst-case running time is poor. Attempts to reconcile this difference have used smoothed analysis, in which adversarial instances are perturbed probabilistically. We are interested in the classical model of smoothed analysis for the Euclidean TSP, in which the perturbations are Gaussian. This model was previously used by Manthey \& Veenstra, who obtained smoothed complexity bounds polynomial in $n$, the dimension $d$, and the perturbation strength $\sigma^{-1}$. However, their analysis only works for $d \geq 4$. The only previous analysis for $d \leq 3$ was performed by Englert, R\"oglin \& V\"ocking, who used a different perturbation model which can be translated to Gaussian perturbations. Their model yields bounds polynomial in $n$ and $\sigma^{-d}$, and super-exponential in $d$. As no direct analysis existed for Gaussian perturbations that yields polynomial bounds for all $d$, we perform this missing analysis. Along the way, we improve all existing smoothed complexity bounds for Euclidean 2-opt.

翻译：2-OPT启发式是一种针对旅行商问题（TSP）的简单局部搜索启发式算法。尽管该算法在实践中通常表现良好，但其最坏情况下的运行时间较差。为协调这种差异，研究者采用了平滑分析，其中对抗性实例被概率性地扰动。我们关注的是欧几里得TSP的经典平滑分析模型，其中扰动服从高斯分布。该模型此前由Manthey和Veenstra使用，他们得到了关于变量个数n、维度d以及扰动强度σ⁻¹的多项式平滑复杂度界。然而，他们的分析仅适用于d ≥ 4的情况。对于d ≤ 3，唯一先前的分析由Englert、Röglin和Vöcking完成，他们使用了另一种可转化为高斯扰动的扰动模型。该模型得到的复杂度界关于n和σ⁻ᵈ为多项式，关于d为超指数形式。由于此前不存在针对高斯扰动的、对所有d均能给出多项式界的直接分析，我们完成了这一缺失的分析。在此过程中，我们改进了所有现有的欧几里得2-OPT平滑复杂度界。