The Landscape of Communication Complexity Classes

Abstract

We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between ${\mathsf{P}}$ and ${\mathsf{PSPACE}}$, short of proving lower bounds against classes for which no explicit lower bounds were already known. Our article also serves as an up-to-date survey on the state of structural communication complexity.

Among our new results we show that ${\mathsf{MA} \not\subseteq \mathsf{ZPP}^{\mathsf{NP}[1]}}$, that is, Merlin–Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one ${\mathsf{NP}}$ query. Here the class $\mathsf{ZPP}^{\mathsf{NP}[1]}$ has the property that generalizing it in the slightest ways would make it contain ${\mathsf{AM} \cap \mathsf{coAM}}$, for which it is notoriously open to prove any explicit lower bounds. We also prove that ${\mathsf{US} \not\subseteq \mathsf{ZPP}^{\mathsf{NP}[1]}}$, where ${\mathsf{US}}$ is the class whose canonically complete problem is the variant of set-disjointness where yes-instances are uniquely intersecting. We also prove that ${\mathsf{US} \not\subseteq \mathsf{coDP}}$, where ${\mathsf{DP}}$ is the class of differences of two ${\mathsf{NP}}$ sets. Finally, we explore an intriguing open issue: Are rank-1 matrices inherently more powerful than rectangles in communication complexity? We prove a new separation concerning ${\mathsf{PP}}$ that sheds light on this issue and strengthens some previously known separations.