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

TL;DR

This paper establishes a tighter upper bound on the smoothed complexity of the FLIP algorithm for local Max-Cut and binary Max-CSP, showing it is unlikely for long flip sequences to have small positive improvements.

Contribution

It improves the upper bound on smoothed complexity for FLIP in Max-Cut and Max-CSP, using analysis of flip sequences and their likelihood.

Findings

Smoothed complexity of FLIP for Max-Cut is at most φ n^{O(√log n)}.

Long flip sequences rarely have all small positive improvements.

The bound extends to all binary Max-CSPs.

Abstract

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 shows~that~it 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.