Covering and Packing of Rectilinear Subdivision

TL;DR

This paper investigates NP-hard geometric covering and packing problems on planar subdivisions induced by axis-parallel line segments, providing complexity results and approximation algorithms for specific cases involving rectangular faces.

Contribution

It establishes NP-hardness for several subdivision problems and offers constant-factor approximation algorithms for the Stabbing-Subdivision problem.

Findings

NP-hardness of covering and packing problems on planar subdivisions

NP-hardness persists even for rectangular faces

Constant factor approximation algorithms for Stabbing-Subdivision

Abstract

We study a class of geometric covering and packing problems for bounded regions on the plane. We are given a set of axis-parallel line segments that induces a planar subdivision with a set of bounded (rectilinear) faces. We are interested in the following problems. (P1) Stabbing-Subdivision: Stab all bounded faces by selecting a minimum number of points in the plane. (P2) Independent-Subdivision: Select a maximum size collection of pairwise non-intersecting bounded faces. (P3) Dominating-Subdivision: Select a minimum size collection of faces such that any other face has a non-empty intersection (i.e., sharing an edge or a vertex) with some selected faces. 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.