Sharp Thresholds Imply Circuit Lower Bounds: from random 2-SAT to Planted Clique
David Gamarnik, Elchanan Mossel, Ilias Zadik
TL;DR
This work establishes a general bridge between sharp threshold phenomena for Boolean functions and average-case circuit lower bounds for bounded-depth circuits. By introducing a debiasing technique, the authors convert p-biased inputs to a constant-bias regime and apply Fourier-analytic tools (Russo–Margulis, LMN extension) to derive explicit depth- and size-dependent lower bounds for circuits that decide properties like k-clique in ER graphs and random 2-SAT, near their thresholds. They further extend the framework to statistical estimation problems via the All-or-Nothing phenomenon, proving that AoN implies strong circuit lower bounds for dense planted-clique variants. A partial converse is proved for monotone graph properties, showing that non-sharp-threshold properties admit efficient AC0 implementations on average. Collectively, the results provide a versatile, general mechanism to certify average-case hardness across combinatorial and statistical problems and illuminate the computational landscape around sharp phase transitions.
Abstract
We show that sharp thresholds for Boolean functions directly imply average-case circuit lower bounds. More formally we show that any Boolean function exhibiting a sharp enough threshold at \emph{arbitrary} critical density cannot be computed by Boolean circuits of bounded depth and polynomial size. We also prove a partial converse: if a monotone graph invariant Boolean function does not have a sharp threshold then it can be computed on average by a Boolean circuit of bounded depth and polynomial size. Our general result also implies new average-case bounded depth circuit lower bounds in a variety of settings. (a) ($k$-cliques) For $k=Θ(n)$, we prove that any circuit of depth $d$ deciding the presence of a size $k$ clique in a random graph requires exponential-in-$n^{Θ(1/d)}$ size. (b)(random 2-SAT) We prove that any circuit of depth $d$ deciding the satisfiability of a random 2-SAT formula requires exponential-in-$n^{Θ(1/d)}$ size. To the best of our knowledge, this is the first bounded depth circuit lower bound for random $k$-SAT for any value of $k \geq 2.$ Our results also provide the first rigorous lower bound in agreement with a conjectured, but debated, "computational hardness" of random $k$-SAT around its satisfiability threshold. (c)(Statistical estimation -- planted $k$-clique) Over the recent years, multiple statistical estimation problems have also been proven to exhibit a "statistical" sharp threshold, called the All-or-Nothing (AoN) phenomenon. We show that AoN also implies circuit lower bounds for statistical problems. As a simple corollary of that, we prove that any circuit of depth $d$ that solves to information-theoretic optimality a "dense" variant of the celebrated planted $k$-clique problem requires exponential-in-$n^{Θ(1/d)}$ size.