Packing and Covering with Segments

We study three fundamental geometric optimization problems - independent set, piercing set, and dominating set - on sets of axis-parallel segments in the plane. We consider special cases in which the segments are either unit length or they are anchored on an inclined line (a line with slope - 1). When the segments are anchored on both sides, we prove that all three problems are NP-complete (Throughout, we refer to NP-completeness of problems, as the decision versions of the NP-hard optimization problems we consider are all readily seen to be in NP.); NP-completeness was known for the corresponding problems with axis-parallel rectangles anchored on an inclined line (Correa et al. [4], Mudgal and Pandit [9], Pandit [10]). Further, we prove that the dominating set problem with unit segments in the plane is NP-complete. When the input segments are anchored on one side of the inclined line, there are polynomial-time algorithms for the independent set and piercing set problems.