We construct a graph with $n$ vertices where the smoothed runtime of the 3-FLIP algorithm for the 3-Opt Local Max-Cut problem can be as large as $2^{\Omega(\sqrt{n})}$. This provides the first example where a local search algorithm for the Max-Cut problem can fail to be efficient in the framework of smoothed analysis. We also give a new construction of graphs where the runtime of the FLIP algorithm for the Local Max-Cut problem is $2^{\Omega(n)}$ for any pivot rule. This graph is much smaller and has a simpler structure than previous constructions.

翻译：我们构造了一个具有$n$个顶点的图，其中针对3-Opt局部最大割问题的3-FLIP算法的平滑运行时间可高达$2^{\Omega(\sqrt{n})}$。这首次展示了最大割问题的局部搜索算法在平滑分析框架下可能失效的例子。我们还给出了一种新的图构造，其中对于任何枢轴规则，局部最大割问题的FLIP算法的运行时间均为$2^{\Omega(n)}$。该图比先前的构造更小且结构更简单。