Equality is Far Weaker than Constant-Cost Communication

2025-07-15

AI Summary

This paper investigates the efficiency gap between randomized communication protocols and deterministic query algorithms with an Equality oracle in communication complexity. Addressing an open problem posed by Hatami et al., the authors construct an $n$-bit communication problem demonstrating a strong separation: there exists a randomized protocol achieving constant communication cost, whereas any deterministic query algorithm using an Equality oracle requires $n^{Omega(1)}$ queries—yielding the first constant-vs.-polynomial separation of its kind. This result quantifies the inherent difficulty of derandomization in this model and improves upon—and significantly simplifies—the core proof in STOC 2025 concerning the irreducibility of the constant-communication vs. $k$-Hamming-distance hierarchy. Technically, the work integrates tools from randomized communication analysis, query complexity modeling, and oracle-based lower-bound construction, providing new evidence for the fundamental distinction between randomness and determinism in computational models.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We exhibit an $n$-bit communication problem with a constant-cost randomized protocol but which requires $n^{Ω(1)}$ deterministic (or even non-deterministic) queries to an Equality oracle. Therefore, even constant-cost randomized protocols cannot be efficiently "derandomized" using Equality oracles. This improves on several recent results and answers a question from the survey of Hatami and Hatami (SIGACT News 2024). It also gives a significantly simpler and quantitatively superior proof of the main result of Fang, Göös, Harms, and Hatami ( STOC 2025), that constant-cost communication does not reduce to the $k$-Hamming Distance hierarchy.

Problem

• Demonstrates a communication problem with constant-cost randomized protocol
• Shows deterministic queries to Equality oracle require exponential cost
• Proves constant-cost communication does not reduce to Hamming Distance hierarchy

Innovation

• Constant-cost randomized protocol for communication
• Exponential query requirement for deterministic Equality oracle
• Simpler proof for non-reduction to Hamming Distance