Advertisement

On the Parameterized Complexity of Some Optimization Problems Related to Multiple-Interval Graphs

  • Minghui Jiang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6129)

Abstract

We show that for any constant t ≥ 2, k -Independent Set and k -Dominating Set in t-track interval graphs are W[1]-hard. This settles an open question recently raised by Fellows, Hermelin, Rosamond, and Vialette. We also give an FPT algorithm for k -Clique in t-interval graphs, parameterized by both k and t, with running time max { t O(k), 2 O(klogk) } ·poly(n), where n is the number of vertices in the graph. This slightly improves the previous FPT algorithm by Fellows, Hermelin, Rosamond, and Vialette. Finally, we use the W[1]-hardness of k -Independent Set in t-track interval graphs to obtain the first parameterized intractability result for a recent bioinformatics problem called Maximal Strip Recovery (MSR). We show that MSR-d is W[1]-hard for any constant d ≥ 4 when the parameter is either the total length of the strips, or the total number of adjacencies in the strips, or the number of strips in the optimal solution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alcón, L., Cerioli, M.R., de Figueiredo, C.M.H., Gutierrez, M., Meidanis, J.: Tree loop graphs. Discrete Applied Mathematics 155, 686–694 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bafna, V., Narayanan, B., Ravi, R.: Nonoverlapping local alignments (weighted independent sets of axis-parallel rectangles). Discrete Applied Mathematics 71, 41–53 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bar-Yehuda, R., Halldórsson, M.M., Naor, J(S.), Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM Journal on Computing 36, 1–15 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bulteau, L., Fertin, G., Rusu, I.: Maximal strip recovery problem with gaps: hardness and approximation algorithms. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 710–719. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 268–277 (2007)Google Scholar
  6. 6.
    Chen, Z., Fu, B., Jiang, M., Zhu, B.: On recovering syntenic blocks from comparative maps. Journal of Combinatorial Optimization 18, 307–318 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Crochemore, M., Hermelin, D., Landau, G.M., Rawitz, D., Vialette, S.: Approximating the 2-interval pattern problem. Theoretical Computer Science 395, 283–297 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  9. 9.
    Fellows, M.R., Hermelin, D., Rosamond, F., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science 410, 53–61 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gambette, P., Vialette, S.: On restrictions of balanced 2-interval graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 55–65. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Griggs, J.R.: Extremal values of the interval number of a graph, II. Discrete Mathematics 28, 37–47 (1979)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Griggs, J.R., West, D.B.: Extremal values of the interval number of a graph. SIAM Journal on Algebraic and Discrete Methods 1, 1–7 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gyárfás, A., West, D.B.: Multitrack interval graphs. Congressus Numerantium 109, 109–116 (1995)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Jiang, M.: Approximation algorithms for predicting RNA secondary structures with arbitrary pseudoknots. IEEE/ACM Transactions on Computational Biology and Bioinformatics, doi:10.1109/TCBB.2008.109 (to appear)Google Scholar
  15. 15.
    Jiang, M.: Inapproximability of maximal strip recovery. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 616–625. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Jiang, M.: Inapproximability of maximal strip recovery: II (Submitted)Google Scholar
  17. 17.
    Joseph, D., Meidanis, J., Tiwari, P.: Determining DNA sequence similarity using maximum independent set algorithms for interval graphs. In: Nurmi, O., Ukkonen, E. (eds.) SWAT 1992. LNCS, vol. 621, pp. 326–337. Springer, Heidelberg (1992)Google Scholar
  18. 18.
    Trotter Jr., W.T., Harary, F.: On double and multiple interval graphs. Journal of Graph Theory 3, 205–211 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Vialette, S.: On the computational complexity of 2-interval pattern matching problems. Theoretical Computer Science 312, 223–249 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Wang, L., Zhu, B.: On the tractability of maximal strip recovery. In: Proceedings of the 6th Annual Conference on Theory and Applications of Models of Computation (TAMC 2009), pp. 400–409 (2009)Google Scholar
  21. 21.
    West, D.B., Shmoys, D.B.: Recognizing graphs with fixed interval number is NP-complete. Discrete Applied Mathematics 8, 295–305 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Zheng, C., Zhu, Q., Sankoff, D.: Removing noise and ambiguities from comparative maps in rearrangement analysis. IEEE/ACM Transactions on Computational Biology and Bioinformatics 4, 515–522 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Minghui Jiang
    • 1
  1. 1.Department of Computer ScienceUtah State UniversityLoganUSA

Personalised recommendations