Covering and Packing of Rectilinear Subdivision

We study a class of geometric covering and packing problems for bounded closed regions on the plane. We are given a set of axis-parallel line segments that induce a planar subdivision with bounded (rectilinear) faces. We are interested in the following problems. (P1) Stabbing-Subdivision: Stab all closed bounded faces by selecting a minimum number of points in the plane. (P2) Independent-Subdivision: Select a maximum size collection of pairwise non-intersecting closed bounded faces. (P3) Dominating-Subdivision: Select a minimum size collection of bounded faces such that every other face has a non-empty intersection (i.e., sharing an edge or a vertex) with some selected face. We show that these problems are \(\mathsf { NP }\)-hard. We even prove that these problems are \(\mathsf { NP }\)-hard when we concentrate only on the rectangular faces of the subdivision. Further, we provide constant factor approximation algorithms for the Stabbing-Subdivision problem.