We bound the smoothed running time of the FLIP algorithm for local Max-Cut as a function of $\alpha$, the arboricity of the input graph. We show that, with high probability, the following holds (where $n$ is the number of nodes and $\phi$ is the smoothing parameter): 1) When $\alpha = O(\sqrt{\log n})$ FLIP terminates in $\phi poly(n)$ iterations. Previous to our results the only graph families for which FLIP was known to achieve a smoothed polynomial running time were complete graphs and graphs with logarithmic maximum degree. 2) For arbitrary values of $\alpha$ we get a running time of $\phi n^{O(\frac{\alpha}{\log n} + \log \alpha)}$. This improves over the best known running time for general graphs of $\phi n^{O(\sqrt{ \log n })}$ for $\alpha = o(\log^{1.5} n)$. Specifically, when $\alpha = O(\log n)$ we get a significantly faster running time of $\phi n^{O(\log \log n)}$.

翻译：我们以输入图的树状度$\alpha$为参数，刻画了局部Max-Cut问题中FLIP算法的平滑运行时间。我们证明了（以高概率成立，其中$n$为节点数，$\phi$为平滑参数）： 1) 当$\alpha = O(\sqrt{\log n})$时，FLIP算法在$\phi poly(n)$次迭代内终止。此前已知FLIP算法能实现平滑多项式运行时间的图族仅限于完全图和对数最大度图。 2) 对于任意$\alpha$值，我们得到运行时间为$\phi n^{O(\frac{\alpha}{\log n} + \log \alpha)}$。当$\alpha = o(\log^{1.5} n)$时，该结果优于已知一般图的最优运行时间$\phi n^{O(\sqrt{ \log n })}$。特别地，当$\alpha = O(\log n)$时，我们获得了显著更快的运行时间$\phi n^{O(\log \log n)}$。