Document Open Access Logo

Neighborhood Complexity and Kernelization for Nowhere Dense Classes of Graphs

Authors Kord Eickmeyer, Archontia C. Giannopoulou, Stephan Kreutzer, O-joung Kwon, Michal Pilipczuk, Roman Rabinovich, Sebastian Siebertz

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2017.63.pdf
  • Filesize: 0.56 MB
  • 14 pages

Document Identifiers

Author Details

Kord Eickmeyer
Archontia C. Giannopoulou
Stephan Kreutzer
O-joung Kwon
Michal Pilipczuk
Roman Rabinovich
Sebastian Siebertz

Cite AsGet BibTex

Kord Eickmeyer, Archontia C. Giannopoulou, Stephan Kreutzer, O-joung Kwon, Michal Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. Neighborhood Complexity and Kernelization for Nowhere Dense Classes of Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ICALP.2017.63

Abstract

We prove that whenever G is a graph from a nowhere dense graph class C, and A is a subset of vertices of G, then the number of subsets of A that are realized as intersections of A with r-neighborhoods of vertices of G is at most f(r,eps)|A|^(1+eps), where r is any positive integer, eps is any positive real, and f is a function that depends only on the class C. This yields a characterization of nowhere dense classes of graphs in terms of neighborhood complexity, which answers a question posed by [Reidl et al., CoRR, 2016]. As an algorithmic application of the above result, we show that for every fixed integer r, the parameterized Distance-r Dominating Set problem admits an almost linear kernel on any nowhere dense graph class. This proves a conjecture posed by [Drange et al., STACS 2016], and shows that the limit of parameterized tractability of Distance-r Dominating Set on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.
Keywords
  • Graph Structure Theory
  • Nowhere Dense Graphs
  • Parameterized Complexity
  • Kernelization
  • Dominating Set

Metrics

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

References

  1. Hans Adler and Isolde Adler. Interpreting nowhere dense graph classes as a classical notion of model theory. European Journal of Combinatorics, 36:322-330, 2014. Google Scholar
  2. Jochen Alber, Michael R. Fellows, and Rolf Niedermeier. Polynomial-time data reduction for dominating set. J. ACM, 51(3):363-384, 2004. Google Scholar
  3. Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. (Meta) Kernelization. J. ACM, 63(5):44:1-44:69, 2016. Google Scholar
  4. Hervé Brönnimann and Michael T. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete &Computational Geometry, 14(4):463-479, 1995. Google Scholar
  5. A. Chervonenkis and V. Vapnik. Theory of uniform convergence of frequencies of events to their probabilities and problems of search for an optimal solution from empirical data. Automation and Remote Control, 32:207-217, 1971. Google Scholar
  6. Anuj Dawar. Homomorphism preservation on quasi-wide classes. J. Comput. Syst. Sci., 76(5):324-332, 2010. Google Scholar
  7. Anuj Dawar and Stephan Kreutzer. Domination problems in nowhere-dense classes. In FSTTCS 2009, volume 4 of LIPIcs, pages 157-168. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2009. Google Scholar
  8. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate Texts in Mathematics. Springer, 2012. Google Scholar
  9. Pål Grønås Drange, Markus Sortland Dregi, Fedor V. Fomin, Stephan Kreutzer, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, Felix Reidl, Fernando Sánchez Villaamil, Saket Saurabh, Sebastian Siebertz, and Somnath Sikdar. Kernelization and sparseness: the case of Dominating Set. In STACS 2016, volume 47 of LIPIcs, pages 31:1-31:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. See arxiv preprint 1411.4575 for full proofs. Google Scholar
  10. Zdeněk Dvořák. Constant-factor approximation of the domination number in sparse graphs. European Journal of Combinatorics, 34(5):833-840, 2013. Google Scholar
  11. Kord Eickmeyer, Archontia C. Giannopoulou, Stephan Kreutzer, O-joung Kwon, Michal Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. Neighborhood complexity and kernelization for nowhere dense classes of graphs. CoRR, abs/1612.08197, 2016. URL: https://arxiv.org/abs/1612.08197.
  12. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. In SODA 2010, pages 503-510. SIAM, 2010. Google Scholar
  13. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Linear kernels for (Connected) Dominating Set on H-minor-free graphs. In SODA 2012, pages 82-93. SIAM, 2012. Google Scholar
  14. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Linear kernels for (Connected) Dominating Set on graphs with excluded topological subgraphs. In STACS 2013, volume 20 of LIPIcs, pages 92-103. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2013. Google Scholar
  15. Jakub Gajarský, Petr Hliněný, Jan Obdržálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar. Kernelization using structural parameters on sparse graph classes. J. Comput. Syst. Sci., 84:219-242, 2017. Google Scholar
  16. Martin Grohe, Stephan Kreutzer, Roman Rabinovich, Sebastian Siebertz, and Konstantinos Stavropoulos. Colouring and covering nowhere dense graphs. In WG 2015, pages 325-338, 2015. Google Scholar
  17. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. In STOC 2014, pages 89-98. ACM, 2014. Google Scholar
  18. Stephan Kreutzer, Roman Rabinovich, and Sebastian Siebertz. Polynomial kernels and wideness properties of nowhere dense graph classes. In SODA 2017, pages 1533-1545. SIAM, 2017. See arxiv preprint 1608.05637 for full proofs. Google Scholar
  19. Piotr Micek, Patrice Ossona de Mendez, Sang-il Oum, and David R. Wood. Personal communication, 2016. Google Scholar
  20. Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion I. Decompositions. European Journal of Combinatorics, 29(3):760-776, 2008. Google Scholar
  21. Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion II. Algorithmic aspects. European Journal of Combinatorics, 29(3):777-791, 2008. Google Scholar
  22. Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion III. Restricted graph homomorphism dualities. European Journal of Combinatorics, 29(4):1012-1024, 2008. Google Scholar
  23. Jaroslav Nešetřil and Patrice Ossona de Mendez. First order properties on nowhere dense structures. The Journal of Symbolic Logic, 75(03):868-887, 2010. Google Scholar
  24. Jaroslav Nešetřil and Patrice Ossona de Mendez. On nowhere dense graphs. European Journal of Combinatorics, 32(4):600-617, 2011. Google Scholar
  25. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. Google Scholar
  26. Felix Reidl, Fernando Sánchez Villaamil, and Konstantinos Stavropoulos. Characterising bounded expansion by neighbourhood complexity. CoRR, abs/1603.09532, 2016. Google Scholar
  27. Norbert Sauer. On the density of families of sets. Journal of Combinatorial Theory, Series A, 13(1):145-147, 1972. Google Scholar
  28. 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
  29. Xuding Zhu. Colouring graphs with bounded generalized colouring number. Discrete Mathematics, 309(18):5562-5568, 2009. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact