Approximating the rectilinear crossing number

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 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 n2+o(1)-time algorithm that finds a straight-line drawing of any n-vertex graph G with cr—(G)+o(n4) 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.