Lower Bounds and Parameterized Approach for Longest Common Subsequence

  • Xiuzhen Huang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4112)


In this paper, different parameterized versions of the longest common subsequence (LCS) problem are extensively investigated and computational lower bound results are derived based on current research progress in parameterized computation. For example, with the number of sequences as the parameter k, the problem is unlikely to be solvable in time f(k)n o(k), where n is the length of each sequence and f is any recursive function. The lower bound result is asymptotically tight in consideration of the dynamic programming approach of time O(n k ). Computational lower bounds for polynomial-time approximation schemes (PTAS) for the LCS problem are also derived. It is shown that the LCS problem has no PTAS of time f(1/ ε)n o(1/ ε) for any recursive function f, unless all SNP problems are solvable in subexponential time. Compared with former results on this problem, this result has its significance. Finally a parameterized approach for the LCS problem is discussed, which is more efficient than the dynamic programming approach, especially when applied to large scale sequences.


Vertex Cover Parameterized Approach Recursive Function Dynamic Programming Approach Longe Common Subsequence 


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  1. 1.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation, Combinatorial Optimization Problems and Their Approximability Properties. Springer, New York (1999)MATHGoogle Scholar
  2. 2.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hallett, M.T., Wareham, H.T.: Parameterized complexity analysis in computational biology. Computer Applications in the Biosciences 11, 49–57 (1995)Google Scholar
  3. 3.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Wareham, H.T.: The parameterized complexity of sequence alignment and consensus. Theoretical Computer Science 147, 31–54 (1995)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cai, L., Chen, J.: On fixed-parameter tractability and approximability of NP optimization problems. Journal Of Computer and System Sciences 54, 465–474 (1997)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Information Processing Letters 64, 165–171 (1997)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Chao, K.M., Pearson, W.R., Miller, W.: Aligning two sequences within a specific diagonal band. Computer Applications in the Biosciences 8, 481–487 (1992)Google Scholar
  7. 7.
    Chen, J., Chor, B., Fellows, M., Huang, X., Juedes, D., Kanj, I., Xia, G.: Tight lower bounds for parameterized NP-hard problems. Information and Computation 201, 216–231 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chen, J., Huang, X., Kanj, I., Xia, G.: Linear FPT reductions and computational lower bounds. In: Proc. of the 36th ACM Symposium on Theory of Computing, pp. 212–221 (2004)Google Scholar
  9. 9.
    Chen, J., Huang, X., Kanj, I., Xia, G.: W-hardness linear FPT-reductions: structural properties and further applications. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 975–984. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Chen, J., Kanj, I., Jia, W.: Vertex Cover: Further observations and further improvements. Journal of Algorithms 41, 280–301 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)MATHGoogle Scholar
  12. 12.
    Deng, X., Li, G., Li, Z., Ma, B., Wang, L.: A PTAS for distinguishing (sub)string selection. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 740–751. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Deng, X., Li, G., Li, Z., Ma, B., Wang, L.: Genetic design of drugs without side-effects. SIAM Journal on Computing 32, 1073–1090 (2003)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Downey, R.G., Estivill-Castro, V., Fellows, M.R., Prieto, E., Rosamond, F.A.: Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems. Electr. Notes Theor. Comput. Sci. 78 (2003)Google Scholar
  15. 15.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, New York (1999)Google Scholar
  16. 16.
    Fellows, M., Gramm, J., Niedermeier, R.: Parameterized intractability of motif search problems. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 262–273. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  17. 17.
    Hallett, M.: An Integrated Complexity Analysi of Problems for Computational Biology, Ph.D. Thesis, University of Victoria (1996)Google Scholar
  18. 18.
    Huang, X.: Parameterized Complexity and Polynomial-time Approximation Schemes, Ph.D. Dissertation, Texas A&M University (2004)Google Scholar
  19. 19.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? Journal Of Computer and System Sciences 63, 512–530 (2001)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Jiang, T., Li, M.: On the approximation of shortest common supersequence and longest common subsequences. SIAM Journal on Computing 24, 1112–1139 (1995)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Li, M., Ma, B., Wang, L.: On the closest string and substring problems. Journal of the ACM 49, 157–171 (2002)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal Of Computer and System Sciences 43, 425–440 (1991)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Papadimitriou, C., Yannakakis, M.: On limited nondeterminism and the complexity of VC dimension. Journal Of Computer and System Sciences 53, 161–170 (1996)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Papadimitriou, C., Yannakakis, M.: On the complexity of database queries. Journal Of Computer and System Sciences 58, 407–427 (1999)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Pietrzak, K.: On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. Journal Of Computer and System Sciences 67, 757–771 (2003)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sze, S.-H.: Lectures notes of Special Topics in Computational Biology, Texas A & M University (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Xiuzhen Huang
    • 1
  1. 1.Department of Computer ScienceArkansas State UniversityArkansasUSA

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