Optimal Monotone Depth-Three Circuit Lower Bounds for Majority

2026-01-07

arXiv.org

AI Summary

This work addresses the long-standing absence of tight size lower bounds for the majority function in monotone depth-3 circuits with bottom fan-in at most 3. By resolving the local enumeration problem Enum(3, t) through an optimal algorithm in the setting of monotone formulas, the paper establishes the first optimal size lower bound for majority in this circuit model. The approach integrates techniques from local enumeration, analysis of monotone Boolean formulas, and circuit complexity theory, yielding tight lower bounds for all $ t \leq n/2 $. This result overcomes the limitations of prior non-tight bounds and constitutes a theoretical breakthrough for this restricted yet fundamental circuit class.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• local enumeration
• monotone circuits
• depth-3 circuits
• Majority function
• circuit lower bounds

Innovation

• monotone circuits
• depth-3 lower bounds
• local enumeration
• Majority function
• optimal algorithm