On the time complexity of rectangular covering problems in the discrete plane

Abstract

This paper addresses the computational complexity of optimization problems dealing with the covering of points in the discrete plane by rectangles. Particularly we prove the NP-hardness of such a problem(class) defined by the following objective function: Simultaneously minimize the total area, the total circumference and the number of rectangles used for covering (where the length of every rectangle side is required to lie in a given interval). By using a tiling argument we also prove that a variant of this problem, fixing only the minimal side length of rectangles, is NP-hard. Such problems may appear at the core of applications like data compression, image processing or numerically solving partial differential equations by multigrid computations. © Springer-Verlag Berlin Heidelberg 2004.