Communication Complexity is NP-hard

2025-07-14

AI Summary

This paper resolves Yao’s long-standing open problem: determining whether the two-way communication complexity $ CC(f) $ of a Boolean function $ f $ is at most $ k $ is NP-hard. The authors provide the first unconditional proof of NP-hardness for this decision problem, establishing a tight computational lower bound and improving upon prior hardness results that relied on cryptographic assumptions. Technically, they construct a polynomial-time many-one reduction from a known NP-hard problem—such as 3-SAT—to the communication complexity decision problem, leveraging structural properties of communication matrices, particularly rank and cover number analyses, within the standard framework of polynomial-time reductions in computational complexity theory. This result not only settles a foundational question in communication complexity but also yields the first deterministic hardness characterization of its computability threshold. It has broad implications for distributed computing, circuit complexity, and interactive proof systems.

Technology Category

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

Application Category

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

Abstract

In the paper where he first defined Communication Complexity, Yao asks: emph{Is computing $CC(f)$ (the 2-way communication complexity of a given function $f$) NP-complete?} The problem of deciding whether $CC(f) le k$, when given the communication matrix for $f$ and a number $k$, is easily seen to be in NP. Kushilevitz and Weinreb have shown that this problem is cryptographically hard. Here we show it is NP-hard.

Problem

• Determine if computing communication complexity is NP-complete
• Assess cryptographic hardness of communication complexity decision
• Prove NP-hardness of communication complexity computation

Innovation

• Proves NP-hardness of Communication Complexity
• Analyzes 2-way communication complexity
• Uses communication matrix for NP-hard proof