WALCOM 2019: WALCOM: Algorithms and Computation pp 381-393

# Covering and Packing of Rectilinear Subdivision

• Satyabrata Jana
• Supantha Pandit
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)

## 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 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.

## Keywords

Planar subdivision Set cover Independent set Dominating set $$\mathsf { NP }$$-hard $$\mathsf {PTAS}$$

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