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

TL;DR

This paper demonstrates that low degree polynomial algorithms cannot efficiently find satisfying assignments for random $k$-SAT formulas at clause densities close to the known satisfiability threshold, revealing a computational hardness barrier.

Contribution

It establishes a new hardness result for a broad class of algorithms, including message passing and local algorithms, at clause densities near the satisfiability threshold.

Findings

Low degree polynomial algorithms fail at high clause densities near the threshold.

Introduces a new many-way overlap gap property for random $k$-SAT.

First hardness result for algorithms within a constant factor of the Fix algorithm.

Abstract

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 satisfying assignments exist with high probability up to clause density $m/n = 2^k \log 2 - \frac12 (\log 2 + 1) + o_k(1)$, 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? We prove that the class of low degree polynomial algorithms cannot find a satisfying assignment at clause density $(1 + o_k(1)) \kappa^* 2^k \log k / k$ for a universal constant $\kappa^* \approx 4.911$. This class encompasses Fix, message passing algorithms including Belief and Survey Propagation guided decimation (with bounded or mildly growing number of rounds), and local algorithms on the factor graph. This is the first hardness result for any class of algorithms at clause density within a constant factor of that achieved by Fix. Our proof establishes and leverages a new many-way overlap gap property tailored to random $k$-SAT.