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Optimal Monotone Depth-Three Circuit Lower Bounds for Majority

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

TL;DR

An optimal algorithm is given that solves local enumeration on monotone formulas for k = 3 and all $t \le n/2$ and obtains an optimal lower bound on the size of monotone depth-3 circuits with bottom fan-in at most 3 computing 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.

Optimal Monotone Depth-Three Circuit Lower Bounds for Majority

TL;DR

An optimal algorithm is given that solves local enumeration on monotone formulas for k = 3 and all and obtains an optimal lower bound on the size of monotone depth-3 circuits with bottom fan-in at most 3 computing Majority.

Abstract

Gurumuhkani et al. (CCC'24) introduced the local enumeration problem as follows: for a natural number and a parameter , given an -variate -CNF with no satisfying assignment with Hamming weight less than , enumerate all satisfying assignments of Hamming weight exactly . They showed that efficient algorithms for local enumeration yield new -SAT algorithms and depth-3 lower bounds for Majority function. As the first non-trivial case, they gave an algorithm for 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 and all . 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.
Paper Structure (86 sections, 12 theorems, 10 equations)

This paper contains 86 sections, 12 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Monotone CNFs and Transversals
  4. Circuits and Transversals
  5. Constructions for Extremal Threshold CNFs
  6. Cliques
  7. Turán-style constructions
  8. Constructions for $0\leq t\leq \lfloor n/k \rfloor$ and $n-k+1 \leq t\leq n$
  9. Constructions for $n\geq 1, k=3$ and $0\leq t\leq \lfloor \frac{n}{2} \rfloor$
  10. Upper Bounds on $\Phi_0(s,t)$
  11. Outline of the proof
  12. Conclusions
  13. Proof of \ref{['thm:upperbounds']}
  14. Configurations
  15. Proof of the upper bounds for $\Phi_0(s,t)$
  16. ...and 71 more sections

Key Result

Theorem 1.1

GurumukhaniPaturiPudlakSaksTalebanfard_2024_CCC Assume that Enum($k$, $t$) can be solved in (expected) time $b(n, k, t)$. Then

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2: Main result
  • Definition 2.2: Transversals
  • Definition 2.4: $t$-threshold CNFs
  • Definition 2.5: Transversal number
  • Definition 2.7: Extremal $t$-threshold $k$-CNFs
  • Definition 2.8
  • Definition 2.10
  • Definition 3.1: Clique
  • Example 3.3
  • ...and 23 more