Measuring Indifference: Unit Interval Vertex Deletion

  • René van Bevern
  • Christian Komusiewicz
  • Hannes Moser
  • Rolf Niedermeier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

Making a graph unit interval by a minimum number of vertex deletions is NP-hard. The problem is motivated by applications in seriation and measuring indifference between data items. We present a fixed-parameter algorithm based on the iterative compression technique that finds in O((14k + 14)k + 1kn6) time a set of k vertices whose deletion from an n-vertex graph makes it unit interval. Additionally, we show that making a graph chordal by at most k vertex deletions is NP-complete even on {claw,net,tent}-free graphs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Luce, R.D.: Semiorders and a theory of utility discrimination. Econometrica 24, 178–191 (1956)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aleskerov, F., Bouyssou, D., Monjardet, B.: Utility Maximization, Choice and Preference. Studies in Economic Theory, vol. 16. Springer, Heidelberg (2007)MATHGoogle Scholar
  3. 3.
    Roberts, F.S.: Indifference graphs. In: Proof Techniques in Graph Theory, pp. 139–146. Academic Press, New York (1969)Google Scholar
  4. 4.
    Roberts, F.S.: Indifference and seriation. Annals of the New York Academy of Sciences 328, 173–182 (1979)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. SIAM, Philadelphia (1999)CrossRefMATHGoogle Scholar
  6. 6.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. System Sci. 20, 219–230 (1980)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chepoi, V., Fichet, B., Seston, M.: Seriation in the presence of errors: NP-hardness of l  ∞ -fitting Robinson structures to dissimilarity matrices. J. Classification 26, 279–296 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chepoi, V., Seston, M.: Seriation in the presence of errors: A factor 16 approximation algorithm for l  ∞ -fitting Robinson structures to distances. Algorithmica (2009); Available electronicallyGoogle Scholar
  9. 9.
    Van Bevern, R.: The Computational Hardness and Tractability of Restricted Seriation Problems on Inaccurate Data. Diplomarbeit. Institut für Informatik, Friedrich-Schiller-Universität, Jena, Germany (2010)Google Scholar
  10. 10.
    Dom, M., Guo, J., Niedermeier, R.: Approximation and fixed-parameter algorithms for consecutive ones submatrix problems. J.Comput. System Sci. 76, 204–221 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Marx, D.: Chordal deletion is fixed-parameter tractable. Algorithmica 57, 747–768 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Villanger, Y.: Proper interval vertex deletion. In: Proc. 5th IPEC. LNCS, Springer, Heidelberg (December 2010)Google Scholar
  13. 13.
    Reed, B., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32, 299–301 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Guo, J., Moser, H., Niedermeier, R.: Iterative compression for exactly solving NP-hard minimization problems. In: Lerner, J., Wagner, D., Zweig, K.A. (eds.) Algorithmics of Large and Complex Networks. LNCS, vol. 5515, pp. 65–80. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theor. Comp. Sci. 1, 237–267 (1976)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fellows, M.R., Guo, J., Moser, H., Niedermeier, R.: A complexity dichotomy for finding disjoint solutions of vertex deletion problems. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 319–330. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Panda, B.S., Das, S.K.: A linear time recognition algorithm for proper interval graphs. Inf. Process. Lett. 87, 153–161 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fouquet, J.L.: A strengthening of Ben Rebea’s lemma. J. Combin. Theory Ser. B 59, 35–40 (1993)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. A. Springer, Heidelberg (2003)MATHGoogle Scholar
  20. 20.
    Bodlaender, H.L.: Kernelization: New upper and lower bound techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2008. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)Google Scholar
  21. 21.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38, 31–45 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • René van Bevern
    • 1
  • Christian Komusiewicz
    • 1
  • Hannes Moser
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

Personalised recommendations