Optimal Monotone Depth-Three Circuit Lower Bounds for Majority

Mohit Gurumukhani; Daniel Kleber; Ramamohan Paturi; Christopher Rosin; Michael Saks; Navid Talebanfard

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arXiv · 2601.04072

Optimal Monotone Depth-Three Circuit Lower Bounds for Majority

Mohit Gurumukhani·Daniel Kleber·Ramamohan Paturi·Christopher Rosin·Michael Saks·Navid Talebanfard

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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.

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Mohit Gurumukhani, Daniel Kleber, Ramamohan Paturi, Christopher Rosin, Michael Saks, Navid Talebanfard. 2026-01-07. Optimal Monotone Depth-Three Circuit Lower Bounds for Majority. https://arxiv.org/abs/2601.04072

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