Theoretical Computer Science ( IF 0.747 ) Pub Date : 2020-08-04 , DOI: 10.1016/j.tcs.2020.07.038
Satyabrata Jana; Supantha Pandit

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 $\mathsf{NP}$-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.

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