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Distributed Certification for Classes of Dense Graphs

Authors Pierre Fraigniaud, Frédéric Mazoit, Pedro Montealegre, Ivan Rapaport, Ioan Todinca

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Author Details

Pierre Fraigniaud
  • IRIF, Université Paris Cité, CNRS, France
Frédéric Mazoit
  • LaBRI, Université de Bordeaux, France
Pedro Montealegre
  • Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile
Ivan Rapaport
  • DIM-CMM (UMI 2807 CNRS), Universidad de Chile, Santiago, Chile
Ioan Todinca
  • LIFO, Université d'Orléans and INSA Centre-Val de Loire, France

Funding

  • Fraigniaud, Pierre: Additional support for ANR projects QuData and DUCAT.
  • Montealegre, Pedro: This work was supported by Centro de Modelamiento Matemático (CMM), FB210005, BASAL funds for centers of excellence from ANID-Chile, and ANID-FONDECYT 1230599
  • Rapaport, Ivan: This work was supported by Centro de Modelamiento Matemático (CMM), FB210005, BASAL funds for centers of excellence from ANID-Chile, and ANID-FONDECYT 1220142.

Cite As Get BibTex

Pierre Fraigniaud, Frédéric Mazoit, Pedro Montealegre, Ivan Rapaport, and Ioan Todinca. Distributed Certification for Classes of Dense Graphs. In 37th International Symposium on Distributed Computing (DISC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 281, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.DISC.2023.20

Abstract

A proof-labeling scheme (PLS) for a boolean predicate Π on labeled graphs is a mechanism used for certifying the legality with respect to Π of global network states in a distributed manner. In a PLS, a certificate is assigned to each processing node of the network, and the nodes are in charge of checking that the collection of certificates forms a global proof that the system is in a correct state, by exchanging the certificates once, between neighbors only. The main measure of complexity is the size of the certificates. Many PLSs have been designed for certifying specific predicates, including cycle-freeness, minimum-weight spanning tree, planarity, etc. 
In 2021, a breakthrough has been obtained, as a "meta-theorem" stating that a large set of properties have compact PLSs in a large class of networks. Namely, for every MSO₂ property Π on labeled graphs, there exists a PLS for Π with O(log n)-bit certificates for all graphs of bounded tree-depth. This result has been extended to the larger class of graphs with bounded tree-width, using certificates on O(log² n) bits. 
We extend this result even further, to the larger class of graphs with bounded clique-width, which, as opposed to the other two aforementioned classes, includes dense graphs. We show that, for every MSO₁ property Π on labeled graphs, there exists a PLS for Π with O(log² n)-bit certificates for all graphs of bounded clique-width. As a consequence, certifying families of graphs such as distance-hereditary graphs and (induced) P₄-free graphs (a.k.a., cographs) can be done using a PLS with O(log² n)-bit certificates, merely because each of these two classes can be specified in MSO₁. In fact, we show that certifying P₄-free graphs can be done with certificates on O(log n) bits only. This is in contrast to the class of C₄-free graphs (which does not have bounded clique-width) which requires Ω̃(√n)-bit certificates.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • CONGEST
  • Proof Labelling Schemes
  • clique-width
  • MSO

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