Local search algorithms for NP-hard problems such as Max-Cut frequently perform much better in practice than worst-case analysis suggests. Smoothed analysis has proved an effective approach to understanding this: a substantial literature shows that when a small amount of random noise is added to input data, local search algorithms typically run in polynomial or quasi-polynomial time. In this paper, we provide the first example where a local search algorithm for the Max-Cut problem fails to be efficient in the framework of smoothed analysis. Specifically, we construct a graph with $n$ vertices where the smoothed runtime of the 3-FLIP algorithm can be as large as $2^{\Omega(\sqrt{n})}$. Additionally, for the setting without random noise, we give a new construction of graphs where the runtime of the FLIP algorithm is $2^{\Omega(n)}$ for any pivot rule. These graphs are much smaller and have a simpler structure than previous constructions.

翻译：针对诸如最大割问题等NP难问题的局部搜索算法，在实际应用中的表现往往远优于最坏情况分析所预示的结果。平滑分析已被证明是理解这一现象的有效途径：大量文献表明，当对输入数据添加少量随机噪声时，局部搜索算法通常能在多项式或拟多项式时间内运行。本文首次给出了在平滑分析框架下，最大割问题的局部搜索算法未能保持高效性的例证。具体而言，我们构造了一个具有$n$个顶点的图，使得3-FLIP算法的平滑运行时间可达$2^{\Omega(\sqrt{n})}$量级。此外，在无随机噪声的设置下，我们针对任意枢轴规则，给出了使FLIP算法运行时间达到$2^{\Omega(n)}$的新图构造。相较于先前的构造，这些图具有更小的规模与更简洁的结构。