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Catalytic Communication

Authors Edward Pyne , Nathan S. Sheffield , William Wang

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Edward Pyne
  • MIT, Cambridge, MA, USA
Nathan S. Sheffield
  • MIT, Cambridge, MA, USA
William Wang
  • MIT, Cambridge, MA, USA

Acknowledgements

We thank Ryan Williams for guidance and suggesting the question of catalytic communication complexity, and Carl Schildkraut for helpful discussion about Ruzsa-Szemerédi graphs.

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Edward Pyne, Nathan S. Sheffield, and William Wang. Catalytic Communication. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 79:1-79:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.79

Abstract

The study of space-bounded computation has drawn extensively from ideas and results in the field of communication complexity. Catalytic Computation (Buhrman, Cleve, Koucký, Loff and Speelman, STOC 2013) studies the power of bounded space augmented with a pre-filled hard drive that can be used non-destructively during the computation. Presently, many structural questions in this model remain open. Towards a better understanding of catalytic space, we define a model of catalytic communication complexity and prove new upper and lower bounds.
In our model, Alice and Bob share a blackboard with a tiny number of free bits, and a larger section with an arbitrary initial configuration. They must jointly compute a function of their inputs, communicating only via the blackboard, and must always reset the blackboard to its initial configuration. We prove several upper and lower bounds:  
1) We characterize the simplest nontrivial model, that of one bit of free space and three rounds, in terms of 𝔽₂ rank. In particular, we give natural problems that are solvable with a minimal-sized blackboard that require near-maximal (randomized) communication complexity, and vice versa. 
2) We show that allowing constantly many free bits, as opposed to one, allows an exponential improvement on the size of the blackboard for natural problems. To do so, we connect the problem to existence questions in extremal graph theory. 
3) We give tight connections between our model and standard notions of non-uniform catalytic computation. Using this connection, we show that with an arbitrary constant number of rounds and bits of free space, one can compute all functions in TC⁰.  We view this model as a step toward understanding the value of filled space in computation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
Keywords
  • Catalytic computation
  • Branching programs
  • Communication complexity

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