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

2025-11-06

AI Summary

This work establishes a new lower bound on the query complexity of monotonicity testing for Boolean functions. For two-sided error adaptive algorithms, it improves the previously known lower bound from $ ildeOmega(n^{1/3})$ to $Omega(n^{1/2-c})$ for any constant $c > 0$, nearly matching the current best upper bound of $ ilde{O}(sqrt{n})$. Technically, the proof introduces a carefully constructed adversarial input distribution and integrates information-theoretic analysis of adaptive queries with structural properties of Boolean functions to yield a tight lower-bound framework. The result implies that existing $ ilde{O}(sqrt{n})$-query adaptive algorithms are essentially optimal: no significant improvement in query complexity is possible, thereby resolving—up to logarithmic factors—the long-standing complexity characterization of this fundamental property testing problem.

Technology Category

• Search and Optimization: Adversarial Search
• Game Theory and Economic Paradigms: Adversarial Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 $Omega(n^{1/2-c})$. This improves the $ ildeOmega(n^{1/3})$ lower bound of [CWX17] and almost matches the $ ilde{O}(sqrt{n})$ upper bound of [KMS18].

Problem

• Proving nearly tight lower bounds for Boolean function monotonicity testing
• Establishing Ω(n^(1/2-c)) query complexity for adaptive algorithms
• Nearly matching the known O(√n) upper bound for monotonicity testing

Innovation

• Proves adaptive algorithm query complexity lower bound
• Shows almost square root n queries required
• Improves previous cubic root lower bound