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 $m/n < 2^k \log 2 - \frac12 (\log 2 + 1) + o_k(1)$, 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? 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, 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.

中文翻译：

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低次多项式随机$k$-SAT的算法相变

令 $\Phi$ 是一个带有 $n$ 变量和 $m$ 子句的均匀随机的 $k$-SAT 公式。我们研究了找到令人满意的 $\Phi$ 分配的算法任务。众所周知，在子句密度 $m/n < 2^k \log 2 - \frac12 (\log 2 + 1) + o_k(1)$ 处，存在令人满意的分配的概率很高，而已知的最佳多项式时间算法，Coja-Oghlan 的 Fix 算法，在低得多的子句密度 $(1 - o_k(1)) 2^k \log k / k$ 下找到令人满意的分配。这就提出了一个问题：是否有可能在更高的子句密度下有效地找到令人满意的分配？为了理解随机 $k$-SAT 的算法阈值，我们研究了低次多项式算法，这是一类强大的算法，包括 Fix、Survey Propagation 引导抽取以及消息传递和局部图算法等范式。我们表明低次多项式算法可以在子句密度 $(1 - o_k(1)) 2^k \log k / k$ 处找到令人满意的赋值，匹配 Fix，而不是在子句密度 $(1 + o_k(1) ) \kappa^* 2^k \log k / k$，其中 $\kappa^* \approx 4.911$。这显示了一类算法的随机 $k$-SAT 的第一次急剧（高达常数因子）计算相变。我们的证明建立并利用了针对随机 $k$-SAT 量身定制的新的多路重叠间隙属性。