The 2-opt heuristic is a very simple local search heuristic for the traveling salesperson problem. In practice it usually converges quickly to solutions within a few percentages of optimality. In contrast to this, its running-time is exponential and its approximation performance is poor in the worst case. Englert, R\"oglin, and V\"ocking (Algorithmica, 2014) provided a smoothed analysis in the so-called one-step model in order to explain the performance of 2-opt on d-dimensional Euclidean instances, both in terms of running-time and in terms of approximation ratio. However, translating their results to the classical model of smoothed analysis, where points are perturbed by Gaussian distributions with standard deviation sigma, yields only weak bounds. We prove bounds that are polynomial in n and 1/sigma for the smoothed running-time with Gaussian perturbations. In addition, our analysis for Euclidean distances is much simpler than the existing smoothed analysis. Furthermore, we prove a smoothed approximation ratio of O(log(1/sigma)). This bound is almost tight, as we also provide a lower bound of Omega(log n/ loglog n) for sigma = O(1/sqrt n). Our main technical novelty here is that, different from existing smoothed analyses, we do not separately analyze objective values of the global and local optimum on all inputs (which only allows for a bound of O(1/sigma)), but simultaneously bound them on the same input.

翻译：2-opt启发式是一种非常简单的旅行商问题局部搜索启发式算法。在实践中，它通常能快速收敛到接近最优解（偏差在几个百分点以内）。然而与此相反，其最坏情况下的运行时间是指数级的，且近似性能较差。Englert、Röglin和Vöcking（Algorithmica, 2014）在所谓的一步模型中进行了平滑分析，从运行时间和近似比两方面解释了2-opt算法在d维欧几里得实例上的性能。但将其结果推广到经典平滑分析模型（其中点由标准差为σ的高斯分布扰动）只能得到弱界。我们证明了在高斯扰动下平滑运行时间关于n和1/σ的多项式界。此外，我们对欧几里得距离的分析比现有平滑分析简单得多。同时，我们证明了O(log(1/σ))的平滑近似比。该界几乎是紧的，因为当σ = O(1/√n)时，我们还给出了Ω(log n/ log log n)的下界。本文的主要技术新颖之处在于：与现有平滑分析不同，我们并非在所有输入上分别分析全局最优解和局部最优解的目标值（这只能得到O(1/σ)的界），而是在同一输入上同时约束它们。