A Linear Kernel for the Complementary Maximal Strip Recovery Problem

  • Haitao Jiang
  • Binhai Zhu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7354)


In this paper, we compute the first linear kernel for the complementary problem of Maximal Strip Recovery (CMSR) — a well-known NP-complete problem in computational genomics. Let k be the parameter which represents the size of the solution. The core of the technique is to first obtain a tight 18k bound on the parameterized solution search space, which is done through a mixed global rules and local rules, and via an inverse amortized analysis. Then we apply additional data-reduction rules to obtain a tight 84k kernel for the problem. Combined with the known algorithm using bounded degree search, we obtain the best FPT algorithm for CMSR to this date, running in O(2.36 k k 2 + n 2) time.


Linear Kernel Local Rule Block Graph Solution Search Space Kernelization Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On Problems without Polynomial Kernels (Extended Abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 563–574. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    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
  3. 3.
    Bulteau, L., Fertin, G., Jiang, M., Rusu, I.: Tractability and Approximability of Maximal Strip Recovery. In: Giancarlo, R., Manzini, G. (eds.) CPM 2011. LNCS, vol. 6661, pp. 336–349. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Chen, Z., Fu, B., Jiang, M., Zhu, B.: On recovering syntenic blocks from comparative maps. Journal of Combinatorial Optimization 18(3), 307–318 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Choi, V., Zheng, C., Zhu, Q., Sankoff, D.: Algorithms for the Extraction of Synteny Blocks from Comparative Maps. In: Giancarlo, R., Hannenhalli, S. (eds.) WABI 2007. LNCS (LNBI), vol. 4645, pp. 277–288. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Cook, S.: The complexity of theorem-proving procedures. In: Proceedings of the 3rd ACM Symp. on Theory of Computing (STOC 1971), pp. 151–158 (1971)Google Scholar
  7. 7.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer (1999)Google Scholar
  8. 8.
    Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Proc. 42nd ACM Symp. Theory of Computation (STOC 2010), Cambridge, MA, USA, pp. 251–260 (2010)Google Scholar
  9. 9.
    Fellows, M.: The Lost Continent of Polynomial Time: Preprocessing and Kernelization. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 276–277. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Fernau, H., Fomin, F., Lokshtanov, D., Raible, D., Saurabh, S., Villanger, Y.: Kernel(s) for problems with no kernel: on out-trees with many leaves. In: Proc. 26th Intl. Symp. on Theoretical Aspects of Computer Science (STACS 2009), pp. 421–432 (2009)Google Scholar
  11. 11.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  12. 12.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: Proc. 40th ACM Symp. Theory of Computation (STOC 2008), Victoria, Canada, pp. 133–142 (2008)Google Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)Google Scholar
  14. 14.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38, 31–45 (2007)CrossRefGoogle 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. In: Lee, D.-T., Chen, D.Z., Ying, S. (eds.) FAW 2010. LNCS, vol. 6213, pp. 53–64. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Jiang, H., Li, Z., Lin, G., Wang, L., Zhu, B.: Exact and approximation algorithms for the complementary maximal strip recovery problem. J. of Combinatorial Optimization 23(4), 493–506 (2012)CrossRefGoogle Scholar
  18. 18.
    Jiang, H., Zhang, C., Zhu, B.: Weak Kernels. ECCC Report, TR10-005 (October 2010)Google Scholar
  19. 19.
    Jiang, H., Zhu, B., Zhu, D.: Algorithms for sorting unsigned linear genomes by the DCJ operations. Bioinformatics 27, 311–316 (2011)CrossRefGoogle Scholar
  20. 20.
    Li, Z., Goebel, R., Wang, L., Lin, G.: An Improved Approximation Algorithm for the Complementary Maximal Strip Recovery Problem. In: Atallah, M., Li, X.-Y., Zhu, B. (eds.) FAW-AAIM 2011. LNCS, vol. 6681, pp. 46–57. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Univ. Press (2006)Google Scholar
  22. 22.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, NY (1972)CrossRefGoogle Scholar
  23. 23.
    Wang, L., Zhu, B.: On the tractability of maximal strip recovery. J. of Computational Biology 17(7), 907–914 (2010); Correction 18(1), 129 (2011)CrossRefGoogle Scholar
  24. 24.
    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 2012

Authors and Affiliations

  • Haitao Jiang
    • 1
    • 2
  • Binhai Zhu
    • 3
  1. 1.School of Computer Science and TechnologyShandong UniversityJinanChina
  2. 2.School of Mathematics and System ScienceShandong UniversityJinanChina
  3. 3.Department of Computer ScienceMontana State UniversityBozemanUSA

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