LIPIcs.GD.2024.54.pdf
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We present a deterministic n^(2+o(1))-time algorithm that approximates the crossing number of any graph G of order n up to an additive error of o(n⁴), as well as a randomized polynomial-time algorithm that constructs a drawing of G with cr(G)+o(n⁴) crossings. These results imply a (1+o(1))-approximation algorithm for the crossing number of dense graphs. Our work builds on the machinery used by Fox, Pach and Súk [Fox et al., 2016], who obtained similar results for the rectilinear crossing number.
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