2009/04/21 by Christoph Dürr, Durr, Christoph, Flavio Guíñez +3
Computer Science · #Computational Complexity (cs.CC) #Computational Geometry and Mesh Generation #Data Structures and Algorithms (cs.DS) #Digital Image Processing Techniques #FOS: Computer and information sciences #Medical Image Segmentation Techniques
paper · pdf · doi:10.48550/arxiv.0904.3169
openalex publication_date 2009/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Durr 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.