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 and in expectation, the following holds (where $n$ is the number of nodes and $\phi$ is the smoothing parameter): 1) When $\alpha = O(\log^{1-\delta} n)$ FLIP terminates in $\phi poly(n)$ iterations, where $\delta \in (0,1]$ is an arbitrarily small constant. 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：我们针对局部最大割问题的FLIP算法，以输入图的树密度α为参数，对其平滑运行时间进行了界定。研究证明，以高概率和期望意义下（其中n为节点数，φ为平滑参数）可得：1) 当α = O(log^{1-δ} n)时，FLIP算法在φ poly(n)次迭代内终止，其中δ ∈ (0,1]为任意小常数。此前已知FLIP算法能获得多项式平滑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α值，运行时间为φ n^{O(α/log n + log α)}。当α = o(log^{1.5} n)时，该结果优于已知一般图上的最优运行时间φ n^{O(√log n)}。特别地，当α = O(log n)时，我们得到了显著更快的运行时间φ n^{O(log log n)}。