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.
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