The threshold for SDP-refutation of random regular NAE-3SAT

TL;DR

This paper investigates the effectiveness of SDP and spectral algorithms in refuting random regular NAE-3SAT instances, establishing a sharp threshold at degree 13.5 where these methods transition from failure to success.

Contribution

It identifies a precise degree threshold for SDP and spectral algorithms to efficiently refute random regular NAE-3SAT instances, highlighting a gap where instances are unsatisfiable but hard to refute.

Findings

SDP fails to refute for degrees less than 13.5

Spectral algorithms succeed in refuting for degrees greater than 13.5

Random regular NAE-3SAT becomes easily refutable above degree 8

Abstract

Unlike its cousin 3SAT, the NAE-3SAT (not-all-equal-3SAT) problem has the property that spectral/SDP algorithms can efficiently refute random instances when the constraint density is a large constant (with high probability). But do these methods work immediately above the "satisfiability threshold", or is there still a range of constraint densities for which random NAE-3SAT instances are unsatisfiable but hard to refute? We show that the latter situation prevails, at least in the context of random regular instances and SDP-based refutation. More precisely, whereas a random $d$-regular instance of NAE-3SAT is easily shown to be unsatisfiable (whp) once $d \geq 8$, we establish the following sharp threshold result regarding efficient refutation: If $d < 13.5$ then the basic SDP, even augmented with triangle inequalities, fails to refute satisfiability (whp), if $d > 13.5$ then even the most basic spectral algorithm refutes satisfiability~(whp).