Local max-cut in smoothed polynomial time

TL;DR

This paper proves that, under smoothed analysis, the local max-cut problem can be solved in polynomial time, providing a formal explanation for its empirical ease compared to NP-hardness.

Contribution

It establishes smoothed polynomial time complexity for local max-cut, improving upon previous quasi-polynomial bounds and confirming the problem's practical tractability.

Findings

Smoothed complexity of local max-cut is polynomial.

Confirms local max-cut is easier than NP-hard in smoothed setting.

Provides theoretical proof aligning with empirical observations.

Abstract

In 1988, Johnson, Papadimitriou and Yannakakis wrote that "Practically all the empirical evidence would lead us to conclude that finding locally optimal solutions is much easier than solving NP-hard problems". Since then the empirical evidence has continued to amass, but formal proofs of this phenomenon have remained elusive. A canonical (and indeed complete) example is the local max-cut problem, for which no polynomial time method is known. In a breakthrough paper, Etscheid and R\"oglin proved that the smoothed complexity of local max-cut is quasi-polynomial, i.e., if arbitrary bounded weights are randomly perturbed, a local maximum can be found in $n^{O(\log n)}$ steps. In this paper we prove smoothed polynomial complexity for local max-cut, thus confirming that finding local optima for max-cut is much easier than solving it.