Document Open Access Logo

FO Model Checking of Geometric Graphs

Authors Petr Hlineny, Filip Pokrývka, Bodhayan Roy



PDF
Thumbnail PDF

File

LIPIcs.IPEC.2017.19.pdf
  • Filesize: 0.49 MB
  • 12 pages

Document Identifiers

Author Details

Petr Hlineny
Filip Pokrývka
Bodhayan Roy

Cite AsGet BibTex

Petr Hlineny, Filip Pokrývka, and Bodhayan Roy. FO Model Checking of Geometric Graphs. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 19:1-19:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.IPEC.2017.19

Abstract

Over the past two decades the main focus of research into first-order (FO) model checking algorithms has been on sparse relational structures – culminating in the FPT algorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs. On contrary to that, except the case of locally bounded clique-width only little is currently known about FO model checking of dense classes of graphs or other structures. We study the FO model checking problem for dense graph classes definable by geometric means (intersection and visibility graphs). We obtain new nontrivial FPT results, e.g., for restricted subclasses of circular-arc, circle, box, disk, and polygon-visibility graphs. These results use the FPT algorithm by Gajarský et al. for FO model checking of posets of bounded width. We also complement the tractability results by related hardness reductions.
Keywords
  • first-order logic
  • model checking
  • fixed-parameter tractability
  • intersection graphs
  • visibility graphs

Metrics

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

References

  1. H. Adler and I. Adler. Interpreting nowhere dense graph classes as a classical notion of model theory. Eur. J. Comb., 36:322-330, 2014. Google Scholar
  2. A. Bouchet. Reducing prime graphs and recognizing circle graphs. Combinatorica, 7:243-254, 1987. Google Scholar
  3. S. Bova, R. Ganian, and S. Szeider. Model checking existential logic on partially ordered sets. ACM Trans. Comput. Log., 17(2):10:1-10:35, 2016. Google Scholar
  4. H. Breu and D. G. Kirkpatrick. Unit disk graph recognition is NP-hard. Computational Geometry, 9(1-2):3-24, 1998. Google Scholar
  5. B. Courcelle, J. A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. Google Scholar
  6. A. Dawar, M. Grohe, and S. Kreutzer. Locally excluding a minor. In LICS'07, pages 270-279. IEEE Computer Society, 2007. Google Scholar
  7. H. N. de Ridder et al. Information System on Graph Classes and their Inclusions (ISGCI). URL: http://www.graphclasses.org.
  8. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: http://dx.doi.org/10.1007/978-1-4471-5559-1.
  9. Rodney G. Downey, Michael R. Fellows, and Udayan Taylor. The parameterized complexity of relational database queries and an improved characterization of W[1]. In First Conference of the Centre for Discrete Mathematics and Theoretical Computer Science, DMTCS 1996, New Zealand, December, 9-13, 1996, pages 194-213. Springer-Verlag, Singapore, 1996. Google Scholar
  10. Z. Dvořák, D. Kráľ, and R. Thomas. Deciding first-order properties for sparse graphs. In FOCS'10, pages 133-142. IEEE Computer Society, 2010. Google Scholar
  11. S. Eidenbenz. In-approximability of finding maximum hidden sets on polygons and terrains. Computational Geometry: Theory and Applications, 21:139-153, 2002. Google Scholar
  12. M. Frick and M. Grohe. Deciding first-order properties of locally tree-decomposable structures. J. ACM, 48(6):1184-1206, 2001. Google Scholar
  13. J. Gajarský, P. Hliněný, D. Lokshtanov, J. Obdržálek, S. Ordyniak, M. S. Ramanujan, and S. Saurabh. FO model checking on posets of bounded width. In FOCS'15, pages 963-974. IEEE Computer Society, 2015. Full paper arXiv:1504.04115. Google Scholar
  14. J. Gajarský, P. Hliněný, J. Obdržálek, and S. Ordyniak. Faster existential FO model checking on posets. In ISAAC'14, volume 8889 of LNCS, pages 441-451. Springer, 2014. Google Scholar
  15. J. Gajarský, P. Hliněný, J. Obdržálek, D. Lokshtanov, and M. S. Ramanujan. A new perspective on FO model checking of dense graph classes. In LICS '16, pages 176-184. ACM, 2016. Google Scholar
  16. R. Ganian, P. Hliněný, D. Kráľ, J. Obdržálek, J. Schwartz, and J. Teska. FO model checking of interval graphs. Log. Methods Comput. Sci., 11(4:11):1-20, 2015. Google Scholar
  17. S. K. Ghosh. Visibility Algorithms in the Plane. Cambridge University Press, 2007. Google Scholar
  18. S. K. Ghosh, A. Maheshwari, S. P. Pal, S. Saluja, and C. E. Veni Madhavan. Characterizing and recognizing weak visibility polygons. Computational Geometry: Theory and Applications, 3:213-233, 1993. Google Scholar
  19. S. K. Ghosh, T. Shermer, B. K. Bhattacharya, and P. P. Goswami. Computing the maximum clique in the visibility graph of a simple polygon. Journal of Discrete Algorithms, 5:524-532, 2007. Google Scholar
  20. M. Grohe, S. Kreutzer, and S. Siebertz. Deciding first-order properties of nowhere dense graphs. In STOC'14, pages 89-98. ACM, 2014. Google Scholar
  21. P. Heggernes, P. van 't Hof, D. Meister, and Y. Villanger. Induced subgraph isomorphism on proper interval and bipartite permutation graphs. Theoretical Computer Science, 562:252-269, 2015. Google Scholar
  22. D. Marx. Efficient approximation schemes for geometric problems? In Algorithms - ESA 2005, 13th Annual European Symposium, Proceedings, volume 3669 of Lecture Notes in Computer Science, pages 448-459. Springer, 2005. Google Scholar
  23. D. Marx and I. Schlotter. Cleaning interval graphs. Algorithmica, 65(2):275-316, 2013. Google Scholar
  24. R. M. McConnell. Linear-time recognition of circular-arc graphs. Algorithmica, 37(2):93-147, 2003. Google Scholar
  25. J. O'Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, New York, 1987. Google Scholar
  26. D. Seese. Linear time computable problems and first-order descriptions. Math. Structures Comput. Sci., 6(6):505-526, 1996. Google Scholar
  27. T. Shermer. Hiding people in polygons. Computing, 42:109-131, 1989. Google Scholar
  28. M. Yannakakis. The complexity of the partial order dimension problem. SIAM J. Algebraic Discrete Methods, 3:351-358, 1982. 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