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 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 ⊈ ZPPNP[1], that is, Merlin-Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one NP query. Here the class ZPPNP[1] has the property that generalizing it in the slightest ways would make it contain AM ∩ coAM, for which it is notoriously open to prove any explicit lower bounds. We also prove that US ⊈ ZPPNP[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 ⊈ 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.

Publication Title

Leibniz International Proceedings in Informatics, LIPIcs

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