Document Open Access Logo

Crossing Number Is NP-Hard for Constant Path-Width (And Tree-Width)

Authors Petr Hliněný , Liana Khazaliya

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2024.40.pdf
  • Filesize: 1.2 MB
  • 15 pages

Document Identifiers

Author Details

Petr Hliněný
  • Masaryk University, Brno, Czech Republic
Liana Khazaliya
  • Technische Universität Wien, Austria

Funding

Liana Khazaliya acknowledges support from the Austrian Science Fund (FWF) [Y1329] and the European Union’s Horizon 2020 COFUND programme [LogiCS@TUWien, grant agreement No.101034440].

Cite As Get BibTex

Petr Hliněný and Liana Khazaliya. Crossing Number Is NP-Hard for Constant Path-Width (And Tree-Width). In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.40

Abstract

Crossing Number is a celebrated problem in graph drawing. It is known to be NP-complete since the 1980s, and fairly involved techniques were already required to show its fixed-parameter tractability when parameterized by the vertex cover number. In this paper we prove that computing exactly the crossing number is NP-hard even for graphs of path-width 12 (and as a result, for simple graphs of path-width 13 and tree-width 9).
Thus, while tree-width and path-width have been very successful tools in many graph algorithm scenarios, our result shows that general crossing number computations unlikely (under P≠ NP) could be successfully tackled using graph decompositions of bounded width, what has been a "tantalizing open problem" [S. Cabello, Hardness of Approximation for Crossing Number, 2013] till now.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Computational geometry
Keywords
  • Graph Drawing
  • Crossing Number
  • Tree-width
  • Path-width

Metrics

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

Related Versions

References

  1. Therese Biedl, Markus Chimani, Martin Derka, and Petra Mutzel. Crossing number for graphs with bounded pathwidth. Algorithmica, 82(2):355-384, 2020. URL: https://doi.org/10.1007/S00453-019-00653-X.
  2. Sergio Cabello. Hardness of approximation for crossing number. Discrete Comput. Geom., 49(2):348-358, March 2013. URL: https://doi.org/10.1007/S00454-012-9440-6.
  3. Sergio Cabello and Bojan Mohar. Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput., 42(5):1803-1829, 2013. URL: https://doi.org/10.1137/120872310.
  4. Éric Colin de Verdière and Thomas Magnard. An FPT algorithm for the embeddability of graphs into two-dimensional simplicial complexes. In Proceedings of the 29th European Symposium on Algorithms (ESA), pages 32:1-32:17, 2021. See also arXiv:2107.06236. URL: https://arxiv.org/abs/2107.06236.
  5. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  6. Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, 1999. Google Scholar
  7. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. URL: https://doi.org/10.1007/978-3-662-53622-3.
  8. Michael R. Garey and David S. Johnson. Crossing number is NP-complete. SIAM J. Algebr. Discrete Methods, 4(3):312-316, September 1983. URL: https://doi.org/10.1137/0604033.
  9. Martin Grohe. Computing crossing numbers in quadratic time. J. Comput. Syst. Sci., 68(2):285-302, 2004. URL: https://doi.org/10.1016/J.JCSS.2003.07.008.
  10. Petr Hliněný. Crossing number is hard for cubic graphs. J. Comb. Theory, Ser. B, 96(4):455-471, 2006. URL: https://doi.org/10.1016/j.jctb.2005.09.009.
  11. Petr Hliněný. Complexity of anchored crossing number and crossing number of almost planar graphs. CoRR, abs/2306.03490, 2023. URL: https://doi.org/10.48550/arXiv.2306.03490.
  12. Petr Hliněný and Gelasio Salazar. On hardness of the joint crossing number. In ISAAC, volume 9472 of Lecture Notes in Computer Science, pages 603-613. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48971-0_51.
  13. Petr Hliněný and Abhisekh Sankaran. Exact crossing number parameterized by vertex cover. In Daniel Archambault and Csaba D. Tóth, editors, Graph Drawing and Network Visualization - 27th International Symposium, GD 2019, Prague, Czech Republic, September 17-20, 2019, Proceedings, volume 11904 of Lecture Notes in Computer Science, pages 307-319. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-35802-0_24.
  14. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Stefankovic. Crossing numbers and parameterized complexity. In Seok-Hee Hong, Takao Nishizeki, and Wu Quan, editors, Graph Drawing, 15th International Symposium, GD 2007, Sydney, Australia, September 24-26, 2007. Revised Papers, volume 4875 of Lecture Notes in Computer Science, pages 31-36. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-77537-9_6.
  15. Paul D. Seymour and Robin Thomas. Graph searching and a min-max theorem for tree-width. J. Comb. Theory B, 58(1):22-33, 1993. URL: https://doi.org/10.1006/jctb.1993.1027.
  16. Paul Turán. A note of welcome. Journal of Graph Theory, 1(1):7-9, 1977. URL: https://doi.org/10.1002/jgt.3190010105.
  17. Meirav Zehavi. Parameterized analysis and crossing minimization problems. Comput. Sci. Rev., 45:100490, 2022. URL: https://doi.org/10.1016/J.COSREV.2022.100490.
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 About DROPS Imprint Privacy Contact