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

Authors Petr Hliněný , Liana Khazaliya

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

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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.

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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

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].

References

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