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

  • Christoph Dürr
  • Flavio Guiñez
  • Martín Matamala
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5757)


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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Christoph Dürr
    • 1
  • Flavio Guiñez
    • 2
  • Martín Matamala
    • 3
  1. 1.CNRS, LIX (UMR 7161)Ecole PolytechniquePalaiseauFrance
  2. 2.DIMUniversidad de ChileSantiagoChile
  3. 3.DIM and CMM (UMI 2807, CNRS)Universidad de ChileSantiagoChile

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