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Boolean function monotonicity testing requires (almost) $n^{1/2}$ queries

Mark Chen, Xi Chen, Hao Cui, William Pires, Jonah Stockwell

TL;DR

This work resolves a long-standing gap in the query complexity of monotonicity testing by proving that any two-sided adaptive monotonicity tester must use at least $ ilde{Ω}(n^{0.5-c})$ queries for any constant $c>0$, nearly matching the known upper bound of $ ilde{O}( ext{√}n)$. The authors develop a sophisticated multilevel Talagrand construction that builds a complete $2ℓ$-level tree of random terms and clauses, and they formulate a strong oracle and the notion of safe outcomes to facilitate a Yao minimax lower-bound argument. Two main results—the $2ℓ$-level adaptive lower bound and the adaptivity-hierarchy bound—are proved by showing that the final algorithmic knowledge remains safe unless a large, coordinated exploration penetrates the layered construction. The paper also extends the analysis to constant rounds of adaptivity, achieving a tight $ ilde{Ω}( ext{√}n)$ bound, and discusses relative-error testing in sparse settings, indicating the broad reach of the multilevel Talagrand framework for related testing problems.

Abstract

We show that for any constant $c>0$, any (two-sided error) adaptive algorithm for testing monotonicity of Boolean functions must have query complexity $Ω(n^{1/2-c})$. This improves the $\tildeΩ(n^{1/3})$ lower bound of [CWX17] and almost matches the $\tilde{O}(\sqrt{n})$ upper bound of [KMS18].

Boolean function monotonicity testing requires (almost) $n^{1/2}$ queries

TL;DR

This work resolves a long-standing gap in the query complexity of monotonicity testing by proving that any two-sided adaptive monotonicity tester must use at least queries for any constant , nearly matching the known upper bound of . The authors develop a sophisticated multilevel Talagrand construction that builds a complete -level tree of random terms and clauses, and they formulate a strong oracle and the notion of safe outcomes to facilitate a Yao minimax lower-bound argument. Two main results—the -level adaptive lower bound and the adaptivity-hierarchy bound—are proved by showing that the final algorithmic knowledge remains safe unless a large, coordinated exploration penetrates the layered construction. The paper also extends the analysis to constant rounds of adaptivity, achieving a tight bound, and discusses relative-error testing in sparse settings, indicating the broad reach of the multilevel Talagrand framework for related testing problems.

Abstract

We show that for any constant , any (two-sided error) adaptive algorithm for testing monotonicity of Boolean functions must have query complexity . This improves the lower bound of [CWX17] and almost matches the upper bound of [KMS18].

Paper Structure

This paper contains 44 sections, 31 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Technical Overview
  4. belovs2016polynomial: Talagrand Functions
  5. chen2017beyond: Two-level Talagrand Functions
  6. Multilevel Talagrand Functions
  7. Proof Overview of \ref{['thm:2l-layer-2s-adaptive-lb']}
  8. Proof Overview of \ref{['thm:adaptivity-hierarchy']}
  9. Previous Work
  10. Preliminaries
  11. Multilevel Talagrand Functions
  12. Multiplexer Trees and Maps
  13. Multilevel Talagrand Functions
  14. Distributions ${\mathcal{D}_{\text{yes}}}$ and ${\mathcal{D}_{\text{no}}}$
  15. Outcomes of Query Points
  16. ...and 29 more sections

Key Result

Theorem 1

For any constant $c>0$, there exists a constant $\epsilon_c$ such that any two-sided, adaptive algorithm for testing whether an unknown Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is monotone or $\epsilon_c$-far from monotone must make ${\Omega}(n^{0.5-c})$ queries.

Figures (2)

  • Figure 1: A picture of the two-level Talagrand construction from chen2017beyond.
  • Figure 3: A diagram of the knowledge of the algorithm for the set $[n]$ by the end of Round 2 (a). The whole rectangle represents $[n]$, and the shaded areas (not including $T_i, C_{i,j}$ or $T_{i,j,k}$) are the $1$-coordinates. The set $D_0$, which contains the anti dictator variable $\mathbf{s}$ is located. It has size $\Theta(n^{3/4})$ and it is disjoint from $T_i, C_{i,j}$ and $T_{i,j,k}$.

Theorems & Definitions (72)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 5: Lemma 4 in fischer2002monotonicity
  • Definition 6: Adaptivity
  • Definition 7
  • Lemma 8
  • proof
  • Lemma 9
  • proof
  • ...and 62 more