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Complexity Landscape for Local Certification

Authors Nicolas Bousquet , Laurent Feuilloley , Sébastien Zeitoun

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ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • Local certification
  • proof-labeling schemes
  • locally checkable proofs
  • space complexity
  • distributed graph algorithms
  • complexity gap

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Abstract

An impressive recent line of work has charted the complexity landscape of distributed graph algorithms. For many settings, it has been determined which time complexities exist, and which do not (in the sense that no local problem could have an optimal algorithm with that complexity). In this paper, we initiate the study of the landscape for space complexity of distributed graph algorithms. More precisely, we focus on the local certification setting, where a prover assigns certificates to nodes to certify a property, and where the space complexity is measured by the size of the certificates. 
Already for anonymous paths and cycles, we unveil a surprising landscape:  
- There is a gap between complexity O(1) and Θ(log log n) in paths. This is the first gap established in local certification. 
- There exists a property that has complexity Θ(log log n) in paths, a regime that was not known to exist for a natural property. 
- There is a gap between complexity O(1) and Θ(log n) in cycles, hence a gap that is exponentially larger than for paths. 
We then generalize our result for paths to the class of trees. Namely, we show that there is a gap between complexity O(1) and Θ(log log d) in trees, where d is the diameter. We finally describe some settings where there are no gaps at all.
To prove our results we develop a new toolkit, based on various results of automata theory and arithmetic, which is of independent interest.

Cite As Get BibTex

Nicolas Bousquet, Laurent Feuilloley, and Sébastien Zeitoun. Complexity Landscape for Local Certification. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.DISC.2025.18

Author Details

Nicolas Bousquet
  • CNRS, INSA Lyon, UCBL, LIRIS, UMR5205, F-69622 Villeurbanne, France
Laurent Feuilloley
  • CNRS, INSA Lyon, UCBL, LIRIS, UMR5205, F-69622 Villeurbanne, France
Sébastien Zeitoun
  • CNRS, INSA Lyon, UCBL, LIRIS, UMR5205, F-69622 Villeurbanne, France

Funding

This work is supported by the ANR grant ENEDISC (ANR-24-CE48-7768).

Acknowledgements

The authors would like to thank Thomas Colcombet for answering our questions about Chrobak normal form, and the reviewers for useful comments.

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