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 偏差是旅行销售商问题( TSP) 的简单本地搜索偏差。 虽然它通常在实际中表现良好, 但它最坏的运行时间却很差。 试图调和这一差异的努力使用了平滑的分析, 而在分析中, 对抗性实例是不稳定的概率性能。 我们感兴趣的是 Euclidean TSP 的经典平滑分析模型, 其扰动是 Gaussian 。 这个模型以前是由Manem {Veenstra 使用的, 后者以美元、 美元尺寸和 渗透性强度来获得平滑的复杂度界限。 然而, 试图调和这一差异的尝试使用平滑的分析方法, 仅用 $\ geq 4 来计算 。 唯一对 $d\leq 3 的分析是由 Egletrt, “ oglin ⁇ V\" ocking, 使用不同的扰动模型, 可以翻译为 Gaussian perburbation 。 。 。 它们模型以 $ $ $ 美元和 美元 subligalalalalalalalalalal $, $n $, laus lausal $x ex exald, rubisald expald disals pas pas ex fald.