Approximating the Rectilinear Crossing Number

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 straightline segment connecting the corresponding two points. The rectilinear crossing number of a graph G, cr(G), is the minimum number of pairs of crossing edges in any straight-line drawing of G. Determining or estimating cr(G) appears to be a difficult problem, and deciding if cr(G) ≤ k is known to be NP-hard. In fact, the asymptotic behavior of cr(Kn) is still unknown. In this paper, we present a deterministic n-time algorithm that finds a straight-line drawing of any n-vertex graph G with cr(G)+ o(n) 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))cr(G) pairs of crossing edges.