Covering segments with unit squares

TL;DR

This paper investigates various versions of the line segment covering problem with axis-parallel unit squares, providing complexity results, approximation algorithms, and polynomial solutions for special cases, and establishing connections with conflict-free covering problems.

Contribution

It introduces new NP-completeness results, approximation algorithms, and polynomial algorithms for specific cases of segment covering with unit squares, and links these problems to conflict-free covering.

Findings

Some problems are NP-complete.

Constant factor approximation algorithms are provided.

Polynomial time exact algorithms exist for certain variations.

Abstract

We study several variations of line segment covering problem with axis-parallel unit squares in $I\!\!R^2$. 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. 2015 on conflict-free covering problem. Our NP-completeness results hold for more simplified types of objects than those of Arkin et al. 2015.