Approximation Algorithms for Hitting Triangle-Free Sets of Line Segments

Abstract

We present polynomial time constant factor approximations on NP-Complete special instances of the Guarding a Set of Segments(GSS) problem. The input to the GSS problem consists of a set of line segments, and the goal is to find a minimum size hitting set of the given set of line segments. We consider the underlying planar graph on the set of intersection points as vertices and the edge set as pairs of vertices which are adjacent on a line segment. Our results are for the subclass of instances of GSS for which the underlying planar graph has girth at least 4. On this class of instances, we show that an optimum solution to the natural hitting set LP can be rounded to yield a 3-factor approximation to the optimum hitting set. The GSS problem remains NP-Complete on the sub-class of such instances. The main technique, that we believe could be quite general, is to round the hitting set LP optimum for special hypergraphs that we identify.