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 ⊆ 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.