Depth-3 circuit lower bounds and $k$-SAT algorithms are intimately related; the state-of-the-art $\Sigma^k_3$-circuit lower bound and the $k$-SAT algorithm are based on the same combinatorial theorem. In this paper we define a problem which reveals new interactions between the two. Define Enum($k$, $t$) problem as: given an $n$-variable $k$-CNF and an initial assignment $\alpha$, output all satisfying assignments at Hamming distance $t$ from $\alpha$, assuming that there are no satisfying assignments of Hamming distance less than $t$ from $\alpha$. Observe that: an upper bound $b(n, k, t)$ on the complexity of Enum($k$, $t$) implies: - Depth-3 circuits: Any $\Sigma^k_3$ circuit computing the Majority function has size at least $\binom{n}{\frac{n}{2}}/b(n, k, \frac{n}{2})$. - $k$-SAT: There exists an algorithm solving $k$-SAT in time $O(\sum_{t = 1}^{n/2}b(n, k, t))$. A simple construction shows that $b(n, k, \frac{n}{2}) \ge 2^{(1 - O(\log(k)/k))n}$. Thus, matching upper bounds would imply a $\Sigma^k_3$-circuit lower bound of $2^{\Omega(\log(k)n/k)}$ and a $k$-SAT upper bound of $2^{(1 - \Omega(\log(k)/k))n}$. The former yields an unrestricted depth-3 lower bound of $2^{\omega(\sqrt{n})}$ solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum($k$, $t$) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum($3$, $\frac{n}{2}$). We show that the expected running time of our algorithm is $1.598^n$, substantially improving on the trivial bound of $3^{n/2} \simeq 1.732^n$. This already improves $\Sigma^3_3$ lower bounds for Majority function to $1.251^n$. The previous bound was $1.154^n$ which follows from the work of H{\aa}stad, Jukna, and Pudl\'ak (Comput. Complex.'95).

翻译：深度3电路下界与$k$-SAT算法密切相关；当前最优的$\Sigma^k_3$电路下界与$k$-SAT算法均基于同一组合定理。本文定义了一个揭示二者新交互关系的问题：定义Enum($k$, $t$)问题如下：给定一个$n$变量$k$-CNF公式与初始赋值$\alpha$，在假设距离$\alpha$的汉明距离小于$t$的满足赋值不存在的前提下，输出所有距离为$t$的满足赋值。注意到：关于Enum($k$, $t$)复杂度的上界$b(n, k, t)$蕴含：- 深度3电路：任何计算多数函数的$\Sigma^k_3$电路规模至少为$\binom{n}{\frac{n}{2}}/b(n, k, \frac{n}{2})$。- $k$-SAT：存在算法在$O(\sum_{t = 1}^{n/2}b(n, k, t))$时间内求解$k$-SAT。一个简单构造表明$b(n, k, \frac{n}{2}) \ge 2^{(1 - O(\log(k)/k))n}$。因此，匹配的上界将导出$\Sigma^k_3$电路下界$2^{\Omega(\log(k)n/k)}$与$k$-SAT上界$2^{(1 - \Omega(\log(k)/k))n}$。前者可得到无限制深度3电路下界$2^{\omega(\sqrt{n})}$，解决一个长期未决的开放问题；后者则突破超强指数时间假设。本文为Enum($k$, $t$)提出随机算法，并引入新分析思路。我们通过考虑该问题的首个非平凡实例Enum($3$, $\frac{n}{2}$)验证了思路的有效性：算法期望运行时间为$1.598^n$，相较于平凡上界$3^{n/2} \simeq 1.732^n$有显著改进。这已将对多数函数的$\Sigma^3_3$下界提升至$1.251^n$，此前H{\aa}stad、Jukna与Pudl\'ak（Comput. Complex.'95）的工作给出的最优下界为$1.154^n$。