Optimal matrix-segmentation by rectangles

We study the problem of decomposing a nonnegative matrix into a nonnegative combination of 0-1-matrices whose ones form a rectangle such that the sum of the coefficients is minimal. We present for the case of two rows an easy algorithm that provides an optimal solution which is integral if the given matrix is integral. An additional integrality constraint makes the problem more difficult if the matrix has more than two rows. An algorithm that is based on the revised simplex method and uses only very few Gomory cuts yields exact integral solutions for integral matrices of reasonable size in a short time. For matrices of large dimension we propose a special greedy algorithm that provides sufficiently good results in numerical experiments.