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NP难问题（如最大割）的局部搜索算法在实际中的表现通常远优于最坏情况分析所预测的结果。平滑分析已被证明是理解这一现象的有效方法：大量文献表明，当在输入数据中加入少量随机噪声时，局部搜索算法通常能在多项式或拟多项式时间内运行。本文首次给出最大割问题中局部搜索算法在平滑分析框架下失效的实例。具体而言，我们构造了一个包含$n$个顶点的图，其中3-FLIP算法的平滑运行时间可高达$2^{\Omega(\sqrt{n})}$。此外，针对无随机噪声的场景，我们给出了新的图构造方法，使得FLIP算法在任何枢轴规则下的运行时间均为$2^{\Omega(n)}$。这些图比以往的构造规模更小且结构更简单。