Local Max-Cut on Sparse Graphs

We bound the smoothed running time of the FLIP algorithm for local Max-Cut as a function of α, 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 ϕ is the smoothing parameter): 1) When α = O(log^{1-δ} n) FLIP terminates in ϕ poly(n) iterations, where δ ∈ (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 α we get a running time of ϕ n^{O(α/(log n) + log α)}. This improves over the best known running time for general graphs of ϕ n^{O(√{log n})} for α = o(log^{1.5} n). Specifically, when α = O(log n) we get a significantly faster running time of ϕ n^{O(log log n)}.