Document Open Access Logo

VC Density of Set Systems Definable in Tree-Like Graphs

Authors Adam Paszke, Michał Pilipczuk

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2020.78.pdf
  • Filesize: 0.75 MB
  • 13 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • treewidth
  • cliquewidth
  • definable sets
  • Vapnik-Chervonenkis density

Metrics

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

Abstract

We study set systems definable in graphs using variants of logic with different expressive power. Our focus is on the notion of Vapnik-Chervonenkis density: the smallest possible degree of a polynomial bounding the cardinalities of restrictions of such set systems. On one hand, we prove that if phi(x,y) is a fixed CMSO_1 formula and C is a class of graphs with uniformly bounded cliquewidth, then the set systems defined by phi in graphs from C have VC density at most |y|, which is the smallest bound that one could expect. We also show an analogous statement for the case when phi(x,y) is a CMSO_2 formula and C is a class of graphs with uniformly bounded treewidth. We complement these results by showing that if C has unbounded cliquewidth (respectively, treewidth), then, under some mild technical assumptions on C, the set systems definable by CMSO_1 (respectively, CMSO_2) formulas in graphs from C may have unbounded VC dimension, hence also unbounded VC density.

Cite As Get BibTex

Adam Paszke and Michał Pilipczuk. VC Density of Set Systems Definable in Tree-Like Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 78:1-78:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.78

Author Details

Adam Paszke
  • University of Warsaw, Poland
Michał Pilipczuk
  • University of Warsaw, Poland

Funding

  • Pilipczuk, Michał: The work on this manuscript is a part of project TOTAL funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 677651).

References

  1. Timothy M. Chan, Elyot Grant, Jochen Könemann, and Malcolm Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pages 1576-1585, 2012. URL: https://doi.org/10.5555/2095116.2095241.
  2. Bruno Courcelle. Monadic second-order definable graph transductions: A survey. Theor. Comput. Sci., 126(1):53-75, 1994. URL: https://doi.org/10.1016/0304-3975(94)90268-2.
  3. Bruno Courcelle. Fly-automata for checking MSO₂ graph properties. Discret. Appl. Math., 245:236-252, 2018. URL: https://doi.org/10.1016/j.dam.2016.10.018.
  4. Bruno Courcelle. From tree-decompositions to clique-width terms. Discret. Appl. Math., 248:125-144, 2018. URL: https://doi.org/10.1016/j.dam.2017.04.040.
  5. Bruno Courcelle and Joost Engelfriet. A logical characterization of the sets of hypergraphs defined by hyperedge replacement grammars. Mathematical Systems Theory, 28(6):515-552, 1995. Google Scholar
  6. 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.
  7. Bruno Courcelle and Sang-il Oum. Vertex-minors, monadic second-order logic, and a conjecture by Seese. J. Comb. Theory, Ser. B, 97(1):91-126, 2007. URL: https://doi.org/10.1016/j.jctb.2006.04.003.
  8. Joost Engelfriet and Vincent van Oostrom. Logical description of contex-free graph languages. J. Comput. Syst. Sci., 55(3):489-503, 1997. URL: https://doi.org/10.1006/jcss.1997.1510.
  9. Martin Grohe and György Turán. Learnability and definability in trees and similar structures. Theory Comput. Syst., 37(1):193-220, 2004. URL: https://doi.org/10.1007/s00224-003-1112-8.
  10. Nabil H. Mustafa and Kasturi R. Varadarajan. Epsilon-approximations and epsilon-nets. CoRR, abs/1702.03676, 2017. URL: http://arxiv.org/abs/1702.03676.
  11. Michał Pilipczuk, Sebastian Siebertz, and Szymon Toruńczyk. On the number of types in sparse graphs. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, pages 799-808, 2018. URL: https://doi.org/10.1145/3209108.3209178.
  12. Michael O Rabin. Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society, 141:1-35, 1969. Google Scholar
  13. Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph. J. Comb. Theory, Ser. B, 41(1):92-114, 1986. URL: https://doi.org/10.1016/0095-8956(86)90030-4.
  14. Norbert Sauer. On the density of families of sets. J. Comb. Theory, Ser. A, 13(1):145-147, 1972. URL: https://doi.org/10.1016/0097-3165(72)90019-2.
  15. Detlef Seese. The structure of models of decidable monadic theories of graphs. Ann. Pure Appl. Logic, 53(2):169-195, 1991. URL: https://doi.org/10.1016/0168-0072(91)90054-P.
  16. Saharon Shelah. A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific Journal of Mathematics, 41(1):247-261, 1972. Google Scholar
  17. Vladimir Vapnik and Alexei Ya. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Soviet Mathematics Doklady, 1971. Google Scholar
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact