Document Open Access Logo

Distributed Certification for Classes of Dense Graphs

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

PDF
Thumbnail PDF

File

LIPIcs.DISC.2023.20.pdf
  • Filesize: 0.82 MB
  • 17 pages

Document Identifiers

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 AsGet 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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Alkida Balliu, Gianlorenzo D'Angelo, Pierre Fraigniaud, and Dennis Olivetti. What can be verified locally? J. Comput. Syst. Sci., 97:106-120, 2018. Google Scholar
  2. Alkida Balliu and Pierre Fraigniaud. Certification of compact low-stretch routing schemes. Comput. J., 62(5):730-746, 2019. Google Scholar
  3. Aviv Bick, Gillat Kol, and Rotem Oshman. Distributed zero-knowledge proofs over networks. In 33rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2426-2458, 2022. Google Scholar
  4. Hans L Bodlaender. Nc-algorithms for graphs with small treewidth. In Graph-Theoretic Concepts in Computer Science: International Workshop WG'88 Amsterdam, The Netherlands, June 15-17, 1988 Proceedings 14, pages 1-10. Springer, 1989. Google Scholar
  5. Richard B. Borie, R. Gary Parker, and Craig A. Tovey. Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica, 7(5&6):555-581, 1992. URL: https://doi.org/10.1007/BF01758777.
  6. Nicolas Bousquet, Laurent Feuilloley, and Théo Pierron. Local certification of graph decompositions and applications to minor-free classes. In 25th International Conference on Principles of Distributed Systems (OPODIS), volume 217 of LIPIcs, pages 22:1-22:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  7. Zvika Brakerski and Boaz Patt-Shamir. Distributed discovery of large near-cliques. Distributed Comput., 24(2):79-89, 2011. Google Scholar
  8. Andreas Brandstädt, Van Bang Le, and Jeremy P. Spinrad. Graph Classes: A Survey. Society for Industrial and Applied Mathematics, 1999. Google Scholar
  9. Keren Censor-Hillel, Eldar Fischer, Gregory Schwartzman, and Yadu Vasudev. Fast distributed algorithms for testing graph properties. Distributed Comput., 32(1):41-57, 2019. Google Scholar
  10. Keren Censor-Hillel, Orr Fischer, Tzlil Gonen, François Le Gall, Dean Leitersdorf, and Rotem Oshman. Fast Distributed Algorithms for Girth, Cycles and Small Subgraphs. In 34th International Symposium on Distributed Computing (DISC), volume 179 of LIPIcs, pages 33:1-33:17. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  11. Keren Censor-Hillel, Ami Paz, and Mor Perry. Approximate proof-labeling schemes. In 24th International Colloquium on Structural Information and Communication Complexity (SIROCCO), LNCS 10641, pages 71-89. Springer, 2017. Google Scholar
  12. Derek G. Corneil and Udi Rotics. On the relationship between clique-width and treewidth. SIAM Journal on Computing, 34(4):825-847, 2005. Google Scholar
  13. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  14. Bruno Courcelle and Joost Engelfriet. Graph structure and monadic second-order logic. A language-theoretic approach. Encyclopedia of Mathematics and its applications, Vol. 138. Cambridge University Press, June 2012. Collection Encyclopedia of Mathematics and Applications, Vol. 138. URL: https://hal.science/hal-00646514.
  15. Bruno Courcelle and Mamadou Moustapha Kanté. Graph operations characterizing rank-width and balanced graph expressions. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 66-75. Springer, 2007. Google Scholar
  16. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
  17. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discret. Appl. Math., 101(1-3):77-114, 2000. URL: https://doi.org/10.1016/S0166-218X(99)00184-5.
  18. Pierluigi Crescenzi, Pierre Fraigniaud, and Ami Paz. Trade-offs in distributed interactive proofs. In 33rd International Symposium on Distributed Computing (DISC), volume 146 of LIPIcs, pages 13:1-13:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  19. Konrad K. Dabrowski and Daniël Paulusma. Clique-width of graph classes defined by two forbidden induced subgraphs. The Computer Journal, 59(5):650-666, 2016. Google Scholar
  20. Andrew Drucker, Fabian Kuhn, and Rotem Oshman. On the power of the congested clique model. In 33rd ACM Symposium on Principles of Distributed Computing (PODC), pages 367-376, 2014. Google Scholar
  21. Andrew Drucker, Fabian Kuhn, and Rotem Oshman. On the power of the congested clique model. In Proceedings of the 2014 ACM symposium on Principles of distributed computing, pages 367-376, 2014. Google Scholar
  22. Talya Eden, Nimrod Fiat, Orr Fischer, Fabian Kuhn, and Rotem Oshman. Sublinear-time distributed algorithms for detecting small cliques and even cycles. Distributed Computing, 35(3):207-234, 2022. Google Scholar
  23. Yuval Emek and Yuval Gil. Twenty-two new approximate proof labeling schemes. In 34th International Symposium on Distributed Computing (DISC), volume 179 of LIPIcs, pages 20:1-20:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  24. Yuval Emek, Yuval Gil, and Shay Kutten. Locally restricted proof labeling schemes. In 36th International Symposium on Distributed Computing (DISC), volume 246 of LIPIcs, pages 20:1-20:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  25. Louis Esperet and Benjamin Lévêque. Local certification of graphs on surfaces. Theor. Comput. Sci., 909:68-75, 2022. Google Scholar
  26. Guy Even, Orr Fischer, Pierre Fraigniaud, Tzlil Gonen, Reut Levi, Moti Medina, Pedro Montealegre, Dennis Olivetti, Rotem Oshman, Ivan Rapaport, and Ioan Todinca. Three notes on distributed property testing. In 31st International Symposium on Distributed Computing (DISC), volume 91 of LIPIcs, pages 15:1-15:30. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  27. Laurent Feuilloley, Nicolas Bousquet, and Théo Pierron. What can be certified compactly? compact local certification of MSO properties in tree-like graphs. In 41st ACM Symposium on Principles of Distributed Computing (PODC), pages 131-140, 2022. Google Scholar
  28. Laurent Feuilloley and Pierre Fraigniaud. Error-sensitive proof-labeling schemes. J. Parallel Distributed Comput., 166:149-165, 2022. Google Scholar
  29. Laurent Feuilloley, Pierre Fraigniaud, and Juho Hirvonen. A hierarchy of local decision. Theor. Comput. Sci., 856:51-67, 2021. Google Scholar
  30. Laurent Feuilloley, Pierre Fraigniaud, Juho Hirvonen, Ami Paz, and Mor Perry. Redundancy in distributed proofs. Distributed Comput., 34(2):113-132, 2021. Google Scholar
  31. Laurent Feuilloley, Pierre Fraigniaud, Pedro Montealegre, Ivan Rapaport, Éric Rémila, and Ioan Todinca. Compact distributed certification of planar graphs. Algorithmica, 83(7):2215-2244, 2021. Google Scholar
  32. Laurent Feuilloley and Juho Hirvonen. Local verification of global proofs. In 32nd International Symposium on Distributed Computing, volume 121 of LIPIcs, pages 25:1-25:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  33. Pierre Fraigniaud, François Le Gall, Harumichi Nishimura, and Ami Paz. Distributed quantum proofs for replicated data. In 12th Innovations in Theoretical Computer Science Conference (ITCS), volume 185 of LIPIcs, pages 28:1-28:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  34. Pierre Fraigniaud, Amos Korman, and David Peleg. Towards a complexity theory for local distributed computing. J. ACM, 60(5):35:1-35:26, 2013. Google Scholar
  35. Pierre Fraigniaud, Frédéric Mazoit, Pedro Montealegre, Ivan Rapaport, and Ioan Todinca. Distributed certification for classes of dense graphs, 2023. URL: https://arxiv.org/abs/2307.14292.
  36. Pierre Fraigniaud, Pedro Montealegre, Ivan Rapaport, and Ioan Todinca. A meta-theorem for distributed certification. In 29th International Colloquium on Structural Information and Communication Complexity (SIROCCO), volume 13298 of LNCS, pages 116-134. Springer, 2022. Google Scholar
  37. Pierre Fraigniaud and Dennis Olivetti. Distributed detection of cycles. ACM Trans. Parallel Comput., 6(3):12:1-12:20, 2019. Google Scholar
  38. Pierre Fraigniaud, Boaz Patt-Shamir, and Mor Perry. Randomized proof-labeling schemes. Distributed Comput., 32(3):217-234, 2019. Google Scholar
  39. Pierre Fraigniaud, Ivan Rapaport, Ville Salo, and Ioan Todinca. Distributed testing of excluded subgraphs. In 30th International Symposium on Distributed Computing (DISC), volume 9888 of LNCS, pages 342-356. Springer, 2016. Google Scholar
  40. Markus Frick and Martin Grohe. The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Log., 130(1-3):3-31, 2004. URL: https://doi.org/10.1016/j.apal.2004.01.007.
  41. François Le Gall, Masayuki Miyamoto, and Harumichi Nishimura. Brief announcement: Distributed quantum interactive proofs. In 36th International Symposium on Distributed Computing (DISC), volume 246 of LIPIcs, pages 48:1-48:3. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  42. Mika Göös and Jukka Suomela. Locally checkable proofs in distributed computing. Theory Comput., 12(1):1-33, 2016. Google Scholar
  43. Ö. Johansson. Clique-decomposition, nlc-decomposition, and modular decomposition - relationships and results for random graphs. Congressus Numerantium, 132:39-60, 1998. Google Scholar
  44. Gillat Kol, Rotem Oshman, and Raghuvansh R. Saxena. Interactive distributed proofs. In 37th ACM Symposium on Principles of Distributed Computing (PODC), pages 255-264. ACM, 2018. Google Scholar
  45. Amos Korman and Shay Kutten. Distributed verification of minimum spanning trees. Distributed Comput., 20(4):253-266, 2007. Google Scholar
  46. Amos Korman, Shay Kutten, and David Peleg. Proof labeling schemes. Distributed Comput., 22(4):215-233, 2010. Google Scholar
  47. Moni Naor, Merav Parter, and Eylon Yogev. The power of distributed verifiers in interactive proofs. In 31st ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1096-115. SIAM, 2020. Google Scholar
  48. Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact