The Algorithmic Phase Transition of Random $k$-SAT for Low Degree Polynomials

Let $\Phi$ be a uniformly random $k$-SAT formula with $n$ variables and $m$ clauses. We study the algorithmic task of finding a satisfying assignment of $\Phi$. It is known that a satisfying assignment exists with high probability at clause density $m/n < 2^k \log 2 - \frac{1}{2} (\log 2 + 1) + o_k(1)$, while the best polynomial-time algorithm known, the Fix algorithm of Coja-Oghlan, finds a satisfying assignment at the much lower clause density $(1 - o_k(1)) 2^k \log k / k$. This prompts the question: is it possible to efficiently find a satisfying assignment at higher clause densities? To understand the algorithmic threshold of random $k$-SAT, we study low degree polynomial algorithms, which are a powerful class of algorithms including Fix, Survey Propagation guided decimation (with bounded or mildly growing number of message passing rounds), and paradigms such as message passing and local graph algorithms. We show that low degree polynomial algorithms can find a satisfying assignment at clause density $(1 - o_k(1)) 2^k \log k / k$, matching Fix, and not at clause density $(1 + o_k(1)) \kappa^* 2^k \log k / k$, where $\kappa^* \approx 4.911$. This shows the first sharp (up to constant factor) computational phase transition of random $k$-SAT for a class of algorithms. Our proof establishes and leverages a new many-way overlap gap property tailored to random $k$-SAT.