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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.

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Correspondence to Thomas Watson.

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Göös, M., Pitassi, T. & Watson, T. The Landscape of Communication Complexity Classes. comput. complex. 27, 245–304 (2018). https://doi.org/10.1007/s00037-018-0166-6

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Keywords

  • landscape
  • communication
  • complexity
  • classes

Subject classification

  • 68Q15