On Covering Z-Grid Points by Rectangles

A problem of combinatorial geometry is discussed: Cover a finite set of points lying on an integer grid in the Euclidean plane by regular rectangles such that the total area, circumference and number of rectangles used is minimized. This problem seems to be NP-hard, which is surely the case for related problems concerning covering points arbitrarily distributed in the plane. Treating the case of minimal rectangle side lengths k < λ (grid constant), we propose an exact deterministic algorithm based on set theoretic dynamic programming, which then is improved by exploiting the rectangular and underlying grid structure. We also discuss a variant given by a further parameter p bounding the maximal possible covering cardinality. For this, we are able to find a time bound by a polynomial of degree O(p). Generalizations to arbitrary values of k and arbitrary (finite) space dimensions are possible.