On the Approximability of the Maximum Interval Constrained Coloring Problem

  • Stefan Canzar
  • Khaled Elbassioni
  • Amr Elmasry
  • Rajiv Raman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6507)


In the Maximum Interval Constrained Coloring problem, we are given a set of intervals on a line and a k-dimensional requirement vector for each interval, specifying how many vertices of each of k colors should appear in the interval. The objective is to color the vertices of the line with k colors so as to maximize the total weight of intervals for which the requirement is satisfied. This \(\mathcal{NP}\)-hard combinatorial problem arises in the interpretation of data on protein structure emanating from experiments based on hydrogen/deuterium exchange and mass spectrometry. For constant k, we give a factor \(O(\sqrt{|{\textsc{Opt}}|})\)-approximation algorithm, where Opt is the smallest-cardinality maximum-weight solution. We show further that, even for k = 2, the problem remains APX-hard.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Althaus, E., Canzar, S., Elbassioni, K.M., Karrenbauer, A., Mestre, J.: Approximating the interval constrained coloring problem. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 210–221. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Althaus, E., Canzar, S., Emmett, M.R., Karrenbauer, A., Marshall, A.G., Meyer-Bäse, A., Zhang, H.: Computing h/d-exchange speeds of single residues from data of peptic fragments. In: SAC, pp. 1273–1277 (2008)Google Scholar
  3. 3.
    Byrka, J., Karrebauer, A., Sanità, L.: Hardness of interval constrained coloring. In: Proceedings of the 9th Latin American Theoretical Informatics Symposium, pp. 583–592 (2009)Google Scholar
  4. 4.
    Dilworth, R.: A decomposition theorem for partially ordered sets. Annals of Mathematics 51, 161–166 (1950)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Elbassioni, K.M., Raman, R., Ray, S., Sitters, R.: On the approximability of the maximum feasible subsystem problem with 0/1-coefficients. In: SODA, pp. 1210–1219 (2009)Google Scholar
  6. 6.
    Althaus, E., Canzar, S., Ehrler, C., Emmett, M.R., Karrenbauer, A., Marshall, A.G., Meyer-Bäse, A., Tipton, J., Zhang, H.: Discrete fitting of hydrogen-deuterium-exchange-data of overlapping fragments. In: Proceedings of the 4th International Conference on Bioinformatics & Computational Biology, pp. 23–30 (2009)Google Scholar
  7. 7.
    Canzar, S., Elbassioni, K., Mestre, J.: A polynomial delay algorithm for enumerating approximate solutions to the interval constraint coloring problem. In: ALENEX, pp. 23–33. SIAM, Philadelphia (2010)Google Scholar
  8. 8.
    Komusiewicz, C., Niedermeier, R., Uhlmann, J.: Deconstructing intractability: A case study for interval constrained coloring. In: Kucherov, G., Ukkonen, E. (eds.) CPM 2009. LNCS, vol. 5577, pp. 207–220. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    de Werra, D., Costa, M.C., Picouleau, C., Ries, B.: On the use of graphs in discrete tomography. 4OR 6 (2008) 101–123MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bentz, C., Costa, M.C., de Werra, D., Picouleau, C., Ries, B.: On a graph coloring problem arising from discrete tomography. Networks 51, 256–267 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Albertson, M., Jamison, R., Hedetniemi, S., Locke, S.: The subchromatic number of a graph. Discrete Mathematics 74, 33–49 (1989)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics, vol. 24. Springer, New York (2003)MATHGoogle Scholar
  13. 13.
    Håstad, J.: Some optimal inapproximability results. J. ACM 48, 798–859 (2001)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Stefan Canzar
    • 1
  • Khaled Elbassioni
    • 2
  • Amr Elmasry
    • 2
    • 3
  • Rajiv Raman
    • 4
  1. 1.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany
  3. 3.Datalogisk InstitutUniversity of CopenhagenCopenhagenDenmark
  4. 4.DIMAP and Department of Computer ScienceUniversity of WarwickUK

Personalised recommendations