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Approximating the Rectilinear Crossing Number

  • Jacob Fox
  • János Pach
  • Andrew SukEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

A straight-line drawing of a graph G is a mapping which assigns to each vertex a point in the plane and to each edge a straight-line segment connecting the corresponding two points. The rectilinear crossing number of a graph \(G, \text {cr}(G)\), is the minimum number of pairs of crossing edges in any straight-line drawing of G. Determining or estimating \(\overline{\mathrm{cr}}(G)\) appears to be a difficult problem, and deciding if \(\overline{\mathrm{cr}}(G)\le k\) is known to be NP-hard. In fact, the asymptotic behavior of \(\overline{\mathrm{cr}}(K_n)\) is still unknown.

In this paper, we present a deterministic \(n^{2+o(1)}\)-time algorithm that finds a straight-line drawing of any n-vertex graph G with \(\overline{\mathrm{cr}}(G) + o(n^4)\) pairs of crossing edges. Together with the well-known Crossing Lemma due to Ajtai et al. and Leighton, this result implies that for any dense n-vertex graph G, one can efficiently find a straight-line drawing of G with \((1 + o(1))\overline{\mathrm{cr}}(G)\) pairs of crossing edges.

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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Stanford UniversityStanfordUSA
  2. 2.EPFL, Lausanne, Switzerland and Courant InstituteNewyorkUSA
  3. 3.University of Illinois at ChicagoChicagoUSA

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