Some provably hard crossing number problems

@article{Bienstock1990SomePH,
  title={Some provably hard crossing number problems},
  author={Daniel Bienstock},
  journal={Discrete \& Computational Geometry},
  year={1990},
  volume={6},
  pages={443-459}
}
  • D. Bienstock
  • Published 1 May 1990
  • Mathematics
  • Discrete & Computational Geometry
This paper presents a connection between the problem of drawing a graph with the minimum number of edge crossings, and the theory of arrangements of pseudolines, a topic well-studied by combinatorialists. In particular, we show that any given arrangement can be forced to occur in every minimum crossing drawing of an appropriate graph. Using some recent results of Goodman, Pollack, and Sturmfels, this yields that there exists no polynomial-time algorithm for producing a straight-line drawing of… 

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  • J. PachG. Tóth
  • Mathematics
    Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
  • 1998
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It is proved that the largest of these numbers (the crossing number) cannot exceed, twice the square of the smallest (the odd-crossing number) and the determination of each of these parameters is an NP-hard problem and it is NP-complete in the case of the crossing number and the odd-Crossing number.
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