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, that is, Merlin–Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one NP query. Here the class ZPPNP 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, 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.
Göös, M., Pitassi, T., & Watson, T. (2018). The Landscape of Communication Complexity Classes. Computational Complexity, 27 (2), 245-304. https://doi.org/10.1007/s00037-018-0166-6