Discrete unit square cover problem

Abstract

In this paper, we consider the discrete unit square cover (DUSC) problem as follows: given a set [Formula: see text] of [Formula: see text] points and a set [Formula: see text] of [Formula: see text] axis-aligned unit squares in [Formula: see text], the objective is (i) to check whether the union of the squares in [Formula: see text] covers all the points in [Formula: see text], and (ii) if the answer is yes, then select a minimum cardinality subset [Formula: see text] such that each point in [Formula: see text] is covered by at least one square in [Formula: see text]. For the DUSC problem:(i)we propose a [Formula: see text]-approximation algorithm, where [Formula: see text] is an integer parameter that defines a trade-off between the running time and the approximation factor of the algorithm. The running time of our proposed algorithm is [Formula: see text]. Our solution of the DUSC problem is based on a simple [Formula: see text]-approximation algorithm for the subproblem strip square cover (SSC) problem, where all the points in [Formula: see text] are lying within a horizontal strip of unit height.(ii)we also propose a 2-approximation algorithm, which runs in [Formula: see text] time. The 2-approximation algorithm is based on an algorithm for the subproblem SSC problem. The algorithm for the subproblem is developed using plane sweep and graph search traversal techniques. We also extend this algorithm to get 2-approximation result for the weighted DUSC problem where the squares are assigned weights, and the aim is to choose a subset [Formula: see text] such that each point in [Formula: see text] is covered by at least one square in [Formula: see text] and the sum of the weights of squares in [Formula: see text] is minimized.