Revisiting the cavity-method threshold for random 3-SAT

TL;DR

This paper uses Monte Carlo simulations to analyze the satisfiability threshold for random 3-SAT, suggesting the cavity method may overestimate the threshold for small k, with implications for understanding phase transitions.

Contribution

It provides a detailed Monte Carlo analysis of the 3-SAT threshold and challenges the cavity method's accuracy for small k, proposing different behaviors above and below a critical k.

Findings

Threshold for 3-SAT is at most 4.262 under certain assumptions

Cavity method may overestimate the threshold for small k

Different behaviors observed for k above and below a critical value

Abstract

A detailed Monte Carlo-study of the satisfiability threshold for random 3-SAT has been undertaken. In combination with a monotonicity assumption we find that the threshold for random 3-SAT satisfies $\alpha_3 \leq 4.262$. If the assumption is correct, this means that the actual threshold value for $k=3$ is lower than that given by the cavity method. In contrast the latter has recently been shown to give the correct value for large $k$. Our result thus indicate that there are distinct behaviors for $k$ above and below some critical $k_c$, and the cavity method may provide a correct mean-field picture for the range above $k_c$.