Covering Segments with Unit Squares

Abstract

We study several variations of line segment covering problem with axis-parallel unit squares in IR 2 . A set S of n line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at least one end-point of each segment. The variations depend on the orientation and length of the input segments. We prove some of these problems to be NP-complete, and give constant factor approximation algorithms for those problems. For some variations, we have polynomial time exact algorithms. For the general version of the problem, where the segments are of arbitrary length and orientation, and the squares are given as input, we propose a factor 16 approximation result based on multilevel linear programming relaxation technique, which may be useful for solving some other problems. Further, we show that our problems have connections with the problems studied by Arkin et al.[2] on conflict-free covering problem. Our NP-completeness results hold for more simplified types of objects than those of Arkin et al. [2].

Discriminating Codes in Geometric Setups

“…We now need to solve the Hitting Set problem, where the objective is to choose a minimum number of center points in R 2 , such that each object in O contains at least one of those chosen points. We solve this problem using a technique followed in Acharyya et al (2019) for covering a set of segments using unit squares. A square centered at a point q inside a cell A ∈ A will stab all the segments whose corresponding objects have common intersection A.…”