Gurumuhkani et al. (CCC'24) introduced the local enumeration problem $Enum(k, t)$ as follows: for a natural number $k$ and a parameter $t$, given an $n$-variate $k$-CNF with no satisfying assignment with Hamming weight less than $t(n)$, enumerate all satisfying assignments of Hamming weight exactly $t(n)$. They showed that efficient algorithms for local enumeration yield new $k$-SAT algorithms and depth-3 lower bounds for Majority function. As the first non-trivial case, they gave an algorithm for $k = 3$ which in particular gave a new lower bound on the size of depth-3 circuits with bottom fan-in at most 3 computing Majority. In this paper, we give an optimal algorithm that solves local enumeration on monotone formulas for $k = 3$ and all $t \le n/2$. In particular, we obtain an optimal lower bound on the size of monotone depth-3 circuits with bottom fan-in at most 3 computing Majority.

翻译：Gurumuhkani等人（CCC'24）引入了局部枚举问题$Enum(k, t)$：对于自然数$k$和参数$t$，给定一个$n$元$k$-CNF，且该公式不存在汉明权小于$t(n)$的可满足赋值，要求枚举所有汉明权恰好为$t(n)$的可满足赋值。他们证明了局部枚举的高效算法可产生新的$k$-SAT算法及多数函数的深度三电路下界。作为首个非平凡情形，他们给出了$k = 3$的算法，该算法特别地为计算多数函数的底层扇入至多为3的深度三电路规模提供了新的下界。本文针对$k = 3$且所有$t \le n/2$的情形，给出了解决单调公式局部枚举问题的最优算法。特别地，我们得到了计算多数函数的底层扇入至多为3的单调深度三电路规模的最优下界。