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A Meta-Theorem for Distributed Certification

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Abstract

Distributed certification, whether it be proof-labeling schemes, locally checkable proofs, etc., deals with the issue of certifying the legality of a distributed system with respect to a given boolean predicate. A certificate is assigned to each process in the system by a non-trustable oracle, and the processes are in charge of verifying these certificates, so that two properties are satisfied: completeness, i.e., for every legal instance, there is a certificate assignment leading all processes to accept, and soundness, i.e., for every illegal instance, and for every certificate assignment, at least one process rejects. The verification of the certificates must be fast, and the certificates themselves must be small. A large quantity of results have been produced in this framework, each aiming at designing a distributed certification mechanism for specific boolean predicates. This paper presents a “meta-theorem”, applying to many boolean predicates at once. Specifically, we prove that, for every boolean predicate on graphs definable in the monadic second-order (MSO) logic of graphs, there exists a distributed certification mechanism using certificates on \(O(\log ^2n)\) bits in n-node graphs of bounded treewidth, with a verification protocol involving a single round of communication between neighbors.

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Notes

  1. Treewidth can be viewed as a measure capturing “how close” a graph is from a tree; roughly, a graph of treewidth k can be decomposed by a sequence of cuts, each involving a separator of size O(k).

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Acknowledgements

The authors are thankful to Éric Rémilá for fruitful discussions on certification schemes related to the one considered in this paper.

Funding

This work was supported by Centro de Modelamiento Matemático (CMM), FB210005, BASAL funds for centers of excellence from ANID-Chile (P.M. and I.R.), FONDECYT 1220142 (I.R.) and FONDECYT 1230599 (P.M.).

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Author notes

  1. Pierre Fraigniaud, Pedro Montealegre, Ivan Rapaport and Ioan Todinca contributed equally to this work.

Authors and Affiliations

  1. IRIF, CNRS and Université Paris Cité, Paris, France

    Pierre Fraigniaud

  2. Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibañez, Santiago, Chile

    Pedro Montealegre

  3. DIM-CMM (UMI 2807 CNRS), Universidad de Chile, Santiago, Chile

    Ivan Rapaport

  4. LIFO, Université d’Orléans and INSA Centre-Val de Loire, Orléans, France

    Ioan Todinca

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  1. Pierre Fraigniaud
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  2. Pedro Montealegre
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  3. Ivan Rapaport
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  4. Ioan Todinca
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All authors contributed equally.

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Correspondence to Pedro Montealegre.

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A preliminary version of this work appeared in the proceedings of the 29th International Colloquium on Structural Information and Communication Complexity SIROCCO 2022, held in Paderborn, Germany.

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Fraigniaud, P., Montealegre, P., Rapaport, I. et al. A Meta-Theorem for Distributed Certification. Algorithmica 86, 585–612 (2024). https://doi.org/10.1007/s00453-023-01185-1

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