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

We show that the smoothed complexity of the FLIP algorithm for local Max-Cut is at most $\smash{\phi n^{O(\sqrt{\log n})}}$, where $n$ is the number of nodes in the graph and $\phi$ is a parameter that measures the magnitude of perturbations applied on its edge weights. This improves the previously best upper bound of $\phi n^{O(\log n)}$ by Etscheid and R"{o}glin. Our result is based on an analysis of long sequences of flips, which showsthatit is very unlikely for every flip in a long sequence to incur a positive but small improvement in the cut weight. We also extend the same upper bound on the smoothed complexity of FLIP to all binary Maximum Constraint Satisfaction Problems.

💡 Research Summary

The paper investigates the smoothed complexity of the FLIP local‑search algorithm for the Max‑Cut problem and its extensions to binary Max‑2CSP and general Binary Function Optimization Problems (BFOP). In the smoothed analysis framework, edge weights of an undirected graph are drawn independently from distributions supported on