Reconstructing 3-Colored Grids from Horizontal and Vertical Projections is NP-Hard: A Solution to the 2-Atom Problem in Discrete Tomography

• Published in: SIAM journal on discrete mathematics Vol. 26; no. 1; pp. 330 - 352
• Main Authors: Dürr, Christoph, Guiñez, Flavio, Matamala, Martin
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
• Subjects: Computer Science; Tomography
• ISSN: 0895-4801; 1095-7146
• DOI: 10.1137/100799733

We consider the problem of coloring a grid using $k$ colors with the restriction that each row and each column has a specific number of cells of each color. This problem has been known as the $(k-1)$-atom problem in the discrete tomography community. 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 $k\geqslant 7$ the problem is NP-hard. Afterward Chrobak and Dürr improved this result by proving that it remains NP-hard for $k\geqslant 4$. We close the gap by showing that for $k=3$ colors the problem is already NP-hard. In addition, we give some results on tiling tomography problems.