Some provably hard crossing number problems

TLDR: It is shown that any given arrangement can be forced to occur in every minimum crossing drawing of an appropriate graph, and that there exists no polynomial-time algorithm for producing a straight-line drawing of a graph, which achieves the minimum number of crossings from among all such drawings.

Abstract: 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 a graph, which achieves the minimum number of crossings from among all such drawings. While this result has no bearing on the P versus NP question, it is fairly negative with regard to applications. We also study the problem of drawing a graph with polygonal edges, to achieve the (unrestricted) minimum number of crossings. Here we obtain a tight bound on the smallest number of breakpoints which are required in the polygonal lines.