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

We prove that computing the deterministic communication complexity D(f) of a Boolean function is NP-hard in the standard protocol-tree model, answering, independently and concurrently with Hirahara-Llango-Loff (arXiv:2507.10426), 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:2411.19003); these lemmas can be reused as black boxes in future lower-bound constructions. The instances produced by our reduction admit optimal protocols for self-similar constructions with strong structural properties, giving a flexible framework for the design of reductions showing NP-hardness of deciding the communication complexity of a Boolean matrix. This complements the work by Hirahara, Ilango, and Loff, which establishes NP-hardness in the same model via a different route; our analysis additionally yields reusable structural guarantees and underpins further consequences concerning inapproximability. 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. Furthermore 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.