New Lower Bounds in Merlin-Arthur Communication and Graph Streaming Verification

Authors Prantar Ghosh , Vihan Shah

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Prantar Ghosh
  • Department of Computer Science, Georgetown Univeristy, Washington, D.C., USA
Vihan Shah
  • Department of Computer Science, University of Waterloo, Canada


We are extremely grateful to Sepehr Assadi for many helpful conversations regarding the project. Prantar Ghosh would also like to thank Amit Chakrabarti and Justin Thaler for insightful discussions. Finally, we thank the anonymous reviewers of ITCS 2024 for their many detailed comments and suggestions that helped with improving the presentation of the paper.

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Prantar Ghosh and Vihan Shah. New Lower Bounds in Merlin-Arthur Communication and Graph Streaming Verification. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 53:1-53:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We present novel lower bounds in the Merlin-Arthur (MA) communication model and the related annotated streaming or stream verification model. The MA communication model extends the classical communication model by introducing an all-powerful but untrusted player, Merlin, who knows the inputs of the usual players, Alice and Bob, and attempts to convince them about the output. We focus on the online MA (OMA) model where Alice and Merlin each send a single message to Bob, who needs to catch Merlin if he is dishonest and announce the correct output otherwise. Most known functions have OMA protocols with total communication significantly smaller than what would be needed without Merlin. In this work, we introduce the notion of non-trivial-OMA complexity of a function. This is the minimum total communication required when we restrict ourselves to only non-trivial protocols where Alice sends Bob fewer bits than what she would have sent without Merlin. We exhibit the first explicit functions that have this complexity superlinear - even exponential - in their classical one-way complexity: this means the trivial protocol, where Merlin communicates nothing and Alice and Bob compute the function on their own, is exponentially better than any non-trivial protocol in terms of total communication. These OMA lower bounds also translate to the annotated streaming model, the MA analogue of single-pass data streaming. We show large separations between the classical streaming complexity and the non-trivial annotated streaming complexity (for the analogous notion in this setting) of fundamental problems such as counting distinct items, as well as of graph problems such as connectivity and k-connectivity in a certain edge update model called the support graph turnstile model that we introduce here.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Streaming models
  • Graph Algorithms
  • Streaming
  • Communication Complexity
  • Stream Verification
  • Merlin-Arthur Communication
  • Lower Bounds


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