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We show that the smoothed complexity of the FLIP algorithm for local Max-Cutis at most $\smash{\phi n^{O(\sqrt{\log n})}}$, where $n$ is the number ofnodes in the graph and $\phi$ is a parameter that measures the magnitude ofperturbations applied on its edge weights. This improves the previously bestupper bound of $\phi n^{O(\log n)}$ by Etscheid and R\"{o}glin. Our result isbased on an analysis of long sequences of flips, which shows~that~it is veryunlikely for every flip in a long sequence to incur a positive but smallimprovement in the cut weight. We also extend the same upper bound on thesmoothed complexity of FLIP to all binary Maximum Constraint SatisfactionProblems.

Smoothed complexity of local Max-Cut and binary Max-CSP