Superpolynomial smoothed complexity of 3-FLIP in Local Max-Cut

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.