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Approximability and Parameterized Complexity of Consecutive Ones Submatrix Problems

  • Michael Dom
  • Jiong Guo
  • Rolf Niedermeier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4484)

Abstract

We develop a refinement of a forbidden submatrix characterization of 0/1-matrices fulfilling the Consecutive Ones Property (C1P). This novel characterization finds applications in new polynomial-time approximation algorithms and fixed-parameter tractability results for the problem to find a maximum-size submatrix of a 0/1-matrix such that the submatrix has the C1P. Moreover, we achieve a problem kernelization based on simple data reduction rules and provide several search tree algorithms. Finally, we derive inapproximability results.

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References

  1. 1.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System Sciences 13, 335–379 (1976)MathSciNetMATHGoogle Scholar
  2. 2.
    Dinur, I., Safra, S.: On the hardness of approximating Minimum Vertex Cover. Annals of Mathematics 162(1), 439–485 (2005)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  4. 4.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  6. 6.
    Habib, M., et al.: Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theoretical Computer Science 234(1–2), 59–84 (2000)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Hajiaghayi, M.T.: Consecutive ones property. Manuscript, University Waterloo, Canada (2000)Google Scholar
  8. 8.
    Hajiaghayi, M.T., Ganjali, Y.: A note on the consecutive ones submatrix problem. Information Processing Letters 83(3), 163–166 (2002)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Håstad, J.: Some optimal inapproximability results. Journal of the ACM 48(4), 798–859 (2001)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Hsu, W.-L.: A simple test for the consecutive ones property. Journal of Algorithms 43, 1–16 (2002)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Hsu, W.-L., McConnell, R.M.: PC trees and circular-ones arrangements. Theoretical Computer Science 296(1), 99–116 (2003)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Khot, S., Regev, O.: Vertex Cover might be hard to approximate to within 2 − ε. In: Proc. 18th IEEE Annual Conference on Computational Complexity, pp. 379–386. IEEE Computer Society Press, Los Alamitos (2003)Google Scholar
  13. 13.
    McConnell, R.M.: A certifying algorithm for the consecutive-ones property. In: Proc. 15th ACM-SIAM SODA, pp. 768–777. SIAM, Philadelphia (2004)Google Scholar
  14. 14.
    Meidanis, J., Porto, O., Telles, G.P.: On the consecutive ones property. Discrete Applied Mathmatics 88, 325–354 (1998)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)MATHGoogle Scholar
  16. 16.
    Tan, J., Zhang, L.: The consecutive ones submatrix problem for sparse matrices. To appear in Algorithmica. Preliminary version titled “Approximation algorithms for the consecutive ones submatrix problem on sparse matrices” appeared in Proc. 15th ISAAC, LNCS vol. 3341, pp. 836–846. Springer, Heidelberg (2004)Google Scholar
  17. 17.
    Tucker, A.C.: Matrix characterizations of circular-arc graphs. Pacific Journal of Mathematics 2(39), 535–545 (1971)MathSciNetGoogle Scholar
  18. 18.
    Tucker, A.C.: A structure theorem for the consecutive 1’s property. Journal of Combinatorial Theory (B) 12, 153–162 (1972)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Michael Dom
    • 1
  • Jiong Guo
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 JenaGermany

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