Covering a Set of Points in Multidimensional Space

Abstract Let P = { p 1 , p 2 ,…, p n } be a set of points in d -space. We study the problem of covering with the minimum number of fixed-size orthogonal hypersquares ( CS d for short) all points in P . We present a fast approximation algorithm that generates provably good solutions and an improved polynomial-time approximation scheme for this problem. A variation of the CS d problem is the CR d problem, covering by fixed-size orthogonal hyperrectangles, where the covering of the points is by hyperrectangles with dimensions D 1 , D 2 ,…, D d instead of hypersquares of size D . Another variation is the CD d problem, where we cover the set of points with hyperdiscs of diameter D . Our algorithms can be easily adapted to these problems.