RSB bounds on the maximum cut

TL;DR

This paper advances the understanding of maximum cut sizes in random regular graphs by establishing tighter 2-RSB upper bounds, improving upon previous 1-RSB bounds predicted by physics-based methods.

Contribution

It introduces 2-RSB upper bounds for maximum cut in random regular graphs and refines their parameters to surpass existing 1-RSB bounds.

Findings

Established 2-RSB upper bounds for maximum cut

Refined parameters to improve bounds beyond 1-RSB predictions

Provided tighter bounds closer to the true maximum cut values

Abstract

In the context of random regular graphs, the size of the maximum cut is probably the second most studied graph parameter after the independence ratio. Zdeborov\'a and Boettcher used the cavity method, a non-rigorous statistical physics technique, to predict one-step replica symmetry breaking (1-RSB) formulas. Coja-Ohglan et al. confirmed these predictions as rigorous upper bounds using the interpolation method. While these upper bounds were not expected to be exact, they may be very close to the true values. In this paper, we establish 2-RSB upper bounds and fine-tune their parameters to beat the aforementioned 1-RSB bounds.