Optimal Monotone Depth-Three Circuit Lower Bounds for Majority

TL;DR

This paper presents an optimal algorithm for local enumeration on monotone formulas, leading to the first optimal lower bounds for monotone depth-3 circuits computing Majority with bottom fan-in at most 3.

Contribution

It introduces an optimal algorithm for local enumeration on monotone formulas for k=3 and all t ≤ n/2, establishing tight lower bounds for monotone depth-3 Majority circuits.

Findings

Optimal local enumeration algorithm for monotone formulas

First tight lower bounds for monotone depth-3 Majority circuits

Improves understanding of circuit complexity for Majority

Abstract

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.