Given an n-vertex graph G, a drawing of G in the plane is a mapping of its
vertices into points of the plane, and its edges into continuous curves,
connecting the images of their endpoints. A crossing in such a drawing is a
point where two such curves intersect. In the Minimum Crossing Number problem,
the goal is to find a drawing of G with minimum number of crossings. The value
of the optimal solution, denoted by OPT, is called the graph's crossing number.
This is a very basic problem in topological graph theory, that has received a
significant amount of attention, but is still poorly understood
algorithmically. The best currently known efficient algorithm produces drawings
with O(log2n)(n+OPT) crossings on bounded-degree graphs, while only a
constant factor hardness of approximation is known. A closely related problem
is Minimum Edge Planarization, in which the goal is to remove a
minimum-cardinality subset of edges from G, such that the remaining graph is
planar. Our main technical result establishes the following connection between
the two problems: if we are given a solution of cost k to the Minimum Edge
Planarization problem on graph G, then we can efficiently find a drawing of G
with at most \poly(d)\cdot k\cdot (k+OPT) crossings, where d is the maximum
degree in G. This result implies an O(n\cdot \poly(d)\cdot
\log^{3/2}n)-approximation for Minimum Crossing Number, as well as improved
algorithms for special cases of the problem, such as, for example, k-apex and
bounded-genus graphs