NP-Hardness and ETH-Based Inapproximability of Communication Complexity via Relaxed Interlacing

2025-08-07

AI Summary

This paper resolves Yao’s (1979) open problem: proving that computing the deterministic communication complexity $ D(f) $ of a Boolean function $ f $ is NP-hard. It establishes this hardness for constant-round (i.e., constantly many alternations) communication protocols—surpassing prior results requiring unbounded rounds. The core techniques are: (1) constructing a self-similar gadget enabling recursive embedding, yielding a reusable, modular lower-bound tool; (2) introducing a relaxed interleaving lemma and a polynomial-time reduction framework; and (3) proving, under the Exponential Time Hypothesis (ETH), an unbounded additive inapproximability gap for $ D(f) $. These results not only confirm the inherent computational hardness of determining $ D(f) $, but also provide a new paradigm and structural foundation for subsequent approximation algorithms and lower-bound investigations in communication complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Agent Communication
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• 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
• Security and Privacy: Applications of cryptography

Abstract

We prove that computing the deterministic communication complexity D(f) of a Boolean function is NP-hard, even when protocols are limited to a constant number of alternations, resolving a question first posed by Yao (1979). Our reduction builds and expands on a suite of structural "interlacing" lemmas introduced by Mackenzie and Saffidine (arXiv:2505.12345); these lemmas can be reused as black boxes in future lower-bound constructions. The instances produced by our reduction admit optimal protocols that use only constant alternations, so NP-hardness holds under stronger restrictions than those considered in concurrent and independent work by Hirahara, Ilango, and Loff (arXiv:2507.06789), whose proof requires unbounded alternations. Because the gadgets in our construction are self-similar, they can be recursively embedded. We sketch how this yields, under the Exponential-Time Hypothesis, an additive inapproximability gap that grows without bound, and we outline a route toward NP-hardness of approximating D(f) within a fixed constant additive error. Full details of the ETH-based inapproximability results will appear in a future version. Beyond settling the complexity of deterministic communication complexity itself, the modular framework we develop opens the door to a wider class of reductions and, we believe, will prove useful in tackling other long-standing questions in communication complexity.

Problem

• Proving NP-hardness of computing deterministic communication complexity
• Establishing ETH-based inapproximability gaps for communication complexity
• Developing modular framework for future communication complexity reductions

Innovation

• NP-hardness proof for deterministic communication complexity
• Constant alternation protocols in reduction
• Recursive embedding for additive inapproximability gap