The Landscape of Communication Complexity Classes

We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between P and 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 MA !subseteq ZPP^{NP[1]}, that is, Merlin–Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one NP query. Here the class ZPP^{NP[1]} has the property that generalizing it in the slightest ways would make it contain AM intersect coAM, for which it is notoriously open to prove any explicit lower bounds. We also prove that US !subseteq ZPP^{NP[1]}, where US is the class whose canonically complete problem is the variant of set-disjointness where yes-instances are uniquely intersecting. We also prove that US !subseteq coDP, where DP is the class of differences of two 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 PP that sheds light on this issue and strengthens some previously known separations.

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