Reconstructing Binary Matrices with Neighborhood Constraints: An NP-hard Problem

  • A. Frosini
  • C. Picouleau
  • S. Rinaldi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4992)


This paper deals with the reconstruction of binary matrices having exactly − 1 − 4 − adjacency constraints from the horizontal and vertical projections. Such a problem is shown to be non polynomial by means of a reduction which involves the classic NP-complete problem 3-color. The result is reached by bijectively mapping all the four different cells involved in 3-color into maximal configurations of 0s and 1s which show the adjacency constraint, and which can be merged into a single binary matrix.


Discrete Tomography polynomial time reduction NP- complete Problem 


  1. 1.
    Brunetti, S., Costa, M.C., Frosini, A., Jarray, F., Picouleau, C.: Reconstruction of binary matrics under adjacency constraints. In: Herman, G.T., Kuba, A. (eds.) Advances in Discrete Tomography and its Applications, pp. 125–150. Birkhäuser, Basel (2007)CrossRefGoogle Scholar
  2. 2.
    Chrobak, M., Couperus, P., D"urr, C., Woeginger, G.: A note on tiling under tomographic constraints. Theor. Comp. Sc. 290, 2125–2136 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chrobak, M., Dürr, C.: Reconstructing Polyatomic Structures from X-Rays: NP Completness Proof for three Atoms. Theor. Comp. Sc. 259, 81–98 (2001)zbMATHCrossRefGoogle Scholar
  4. 4.
    Herman, G., Kuba, A.: Discrete Tomography: Foundations, Algorithms and Applications. Birkhauser, Basel (1999)zbMATHGoogle Scholar
  5. 5.
    Herman, G., Kuba, A.: Advances in Discrete Tomography and its Applications. Birkhäuser, Basel (2007)zbMATHCrossRefGoogle Scholar
  6. 6.
    Jarray, F.: Résolution de problèmes de tomographie discrète. Applications à la planification de personnel, Ph. D. thesis, CNAM, Paris, France (2004)Google Scholar
  7. 7.
    Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9, 371–377 (1957)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Ryser, H.J.: Combinatorial mathematics, Mathematical Association of America and Quinn & Boden, Rahway, New Jersey (1963)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • A. Frosini
    • 1
  • C. Picouleau
    • 2
  • S. Rinaldi
    • 3
  1. 1.Dipartimento di Sistemi e InformaticaUniversità di Firenze(Firenze)(Italy)
  2. 2.Laboratoire CEDRIC CNAM(Paris)(France)
  3. 3.Dipartimento di Scienze Matematiche ed InformaticheUniversità di Siena(Siena)(Italy)

Personalised recommendations