On the Tractability of Maximal Strip Recovery

  • Lusheng Wang
  • Binhai Zhu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5532)


Given two genomic maps G and H represented by a sequence of n gene markers, a strip (syntenic block) is a sequence of distinct markers of length at least two which appear as subsequences in the input maps, either directly or in reversed and negated form. The problem Maximal Strip Recovery (MSR) is to find two subsequences G′ and H′ of G and H, respectively, such that the total length of disjoint strips in G′ and H′ is maximized (or, conversely, the number of markers hence deleted, is minimized). Previously, besides some heuristic solutions, a factor-4 polynomial-time approximation is known for the MSR problem; moreover, several close variants of MSR, MSR-d (with d > 2 input maps), MSR-DU (with marker duplications) and MSR-WT (with markers weighted) are all shown to be NP-complete. Before this work, the complexity of the original MSR problem was left open. In this paper, we solve the open problem by showing that MSR is NP-complete, using a polynomial time reduction from One-in-Three 3SAT. We also solve the MSR problem and its variants exactly with FPT algorithms, i.e., showing that MSR is fixed-parameter tractable. Let k be the minimum number of markers deleted in various versions of MSR, the running time of our algorithms are O(22.73k n + n 2) for MSR, O(22.73k dn + dn 2) for MSR-d, and O(25.46k n + n 2) for MSR-DU.


Conjunctive Normal Form Truth Assignment Distinct Marker Syntenic Block Close Variant 
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  1. 1.
    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
  2. 2.
    Blin, G., Fertin, G., Vialette, S.: Extracting constrained 2-interval subsets in 2-interval sets. Theoretical Computer Science 385, 241–263 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen, Z., Fu, B., Jiang, M., Zhu, B.: On recovering syntenic blocks from comparative maps. In: Yang, B., Du, D.-Z., Wang, C.A. (eds.) COCOA 2008. LNCS, vol. 5165, pp. 319–327. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Chen, E., Yang, L., Yuan, H.: Improved algorithms for largest cardinality 2-interval pattern problem. Journal of Combinatorial Optimization 13, 263–275 (2007)zbMATHCrossRefMathSciNetGoogle 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.
    Crochemore, M., Hermelin, D., Landau, G.M., Rawitz, D., Vialette, S.: Approximating the 2-interval pattern problem. Theoretical Computer Science (to appear); preliminary version appeared in Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 426–437. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  9. 9.
    Jiang, M.: A 2-approximation for the preceding-and-crossing structured 2-interval pattern problem. Journal of Combinatorial Optimization 13, 217–221 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jiang, M.: Improved approximation algorithms for predicting RNA secondary structures with arbitrary pseudoknots. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 399–410. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Jiang, M.: A PTAS for the weighted 2-interval pattern problem over the preceding-and-crossing model. In: Dress, A.W.M., Xu, Y., Zhu, B. (eds.) COCOA 2007. LNCS, vol. 4616, pp. 378–387. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Schaefer, T.: The complexity of satisfiability problem. In: Proceedings of the 10th ACM Symposium on Theory of Computing (STOC 1978), pp. 216–226 (1978)Google Scholar
  13. 13.
    Vialette, S.: On the computational complexity of 2-interval pattern matching problems. Theoretical Computer Science 312, 223–249 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    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 2009

Authors and Affiliations

  • Lusheng Wang
    • 1
  • Binhai Zhu
    • 2
  1. 1.Department of Computer ScienceCity University of Hong Kong, KowloonHong Kong
  2. 2.Department of Computer ScienceMontana State UniversityBozemanUSA

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