Covering and packing of rectilinear subdivision

Abstract

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 of the planar subdivision 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 of the planar subdivision. (P3) DOMINATING-SUBDIVISION: Select a minimum size collection of bounded faces of the planar subdivision such that every other face of the subdivision that is not selected has a non-empty intersection (i.e., sharing an edge or a vertex) with some selected face. We show that these problems are NP-hard. We even prove that these problems are 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.