Sharp Thresholds Imply Circuit Lower Bounds: from random 2-SAT to Planted Clique

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 $k=\Theta(n)$, we prove that any circuit of depth $d$ deciding the presence of a size $k$ clique in a random graph requires exponential-in-$n^{\Theta(1/d)}$ 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-$n^{\Theta(1/d)}$ 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-$n^{\Theta(1/d)}$ size.