Reconstructing 3-Colored Grids from Horizontal and Vertical Projections Is NP-hard

We consider the problem of coloring a grid using k colors with the restriction that in each row and each column has an specific number of cells of each color. In an already classical result, Ryser obtained a necessary and sufficient condition for the existence of such a coloring when two colors are considered. This characterization yields a linear time algorithm for constructing such a coloring when it exists. Gardner et al. showed that for k ≥ 7 the problem is NP-hard. Afterward Chrobak and Dürr improved this result, by proving that it remains NP-hard for k ≥ 4. We solve the gap by showing that for 3 colors the problem is already NP-hard. Besides we also give some results on tiling tomography problems.