Extremal Cuts of Sparse Random Graphs

TL;DR

This paper determines the asymptotic sizes of extremal cuts in sparse random graphs, linking their properties to the ground state energy of the Sherrington-Kirkpatrick model through advanced statistical physics techniques.

Contribution

It establishes precise asymptotic formulas for Max-Cut, maximum bisection, and minimum bisection in sparse Erdős-Rényi and regular graphs, connecting graph cuts to spin glass models.

Findings

Max-Cut and maximum bisection sizes are approximately proportional to $rac{ ext{degree}}{4} + P_* imes ext{sqrt}(rac{ ext{degree}}{4})$.

Minimum bisection size is approximately $rac{ ext{degree}}{4} - P_* imes ext{sqrt}(rac{ ext{degree}}{4})$.

The constant $P_*$ is approximately 0.7632, derived from the ground state energy of the Sherrington-Kirkpatrick model.

Abstract

For Erd\H{o}s-R\'enyi random graphs with average degree $\gamma$, and uniformly random $\gamma$-regular graph on $n$ vertices, we prove that with high probability the size of both the Max-Cut and maximum bisection are $n\Big(\frac{\gamma}{4} + {{\sf P}}_* \sqrt{\frac{\gamma}{4}} + o(\sqrt{\gamma})\Big) + o(n)$ while the size of the minimum bisection is $n\Big(\frac{\gamma}{4}-{{\sf P}}_*\sqrt{\frac{\gamma}{4}} + o(\sqrt{\gamma})\Big) + o(n)$. Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington-Kirkpatrick model, with ${{\sf P}}_* \approx 0.7632$ standing for the ground state energy of the latter, expressed analytically via Parisi's formula.