3 research outputs found
On Graph Crossing Number and Edge Planarization
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 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 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
An Algorithm for the Graph Crossing Number Problem
We study the Minimum Crossing Number problem: given an -vertex graph ,
the goal is to find a drawing of in the plane with minimum number of edge
crossings. This is one of the central problems in topological graph theory,
that has been studied extensively over the past three decades. The first
non-trivial efficient algorithm for the problem, due to Leighton and Rao,
achieved an -approximation for bounded degree graphs. This
algorithm has since been improved by poly-logarithmic factors, with the best
current approximation ratio standing on O(n \poly(d) \log^{3/2}n) for graphs
with maximum degree . In contrast, only APX-hardness is known on the
negative side.
In this paper we present an efficient randomized algorithm to find a drawing
of any -vertex graph in the plane with O(OPT^{10}\cdot \poly(d \log
n)) crossings, where is the number of crossings in the optimal solution,
and is the maximum vertex degree in . This result implies an
\tilde{O}(n^{9/10} \poly(d))-approximation for Minimum Crossing Number, thus
breaking the long-standing -approximation barrier for
bounded-degree graphs