Approximating the Obstacle Number for a Graph Drawing Efficiently

An obstacle representation for a (straight-line) graph drawing consists of the positions of the graph vertices together with a set of polygonal obstacles such that every line segment between a pair of non-adjacent vertices intersects some obstacle, while the vertices and edges of the drawing avoid all the obstacles. The obstacle number obs(D) for a graph drawing D is the least number of obstacles in an obstacle representation for it. We present an efficient algorithm for computing the obstacle number for a given graph drawing D with approximation ratio O(log obs(D)). This is achieved by showing that the V-C dimension is bounded for the family of hypergraphs of the underlying transversal problem, and using results from epsilon net theory.