Abstract
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.
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References
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)
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)
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)
Cai, L., Chen, J.: On fixed-parameter tractability and approximability of NP optimization problems. Journal Of Computer and System Sciences 54, 465–474 (1997)
Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Information Processing Letters 64, 165–171 (1997)
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)
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)
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)
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)
Chen, J., Kanj, I., Jia, W.: Vertex Cover: Further observations and further improvements. Journal of Algorithms 41, 280–301 (2001)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)
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)
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)
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)
Downey, R., Fellows, M.: Parameterized Complexity. Springer, New York (1999)
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)
Hallett, M.: An Integrated Complexity Analysi of Problems for Computational Biology, Ph.D. Thesis, University of Victoria (1996)
Huang, X.: Parameterized Complexity and Polynomial-time Approximation Schemes, Ph.D. Dissertation, Texas A&M University (2004)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? Journal Of Computer and System Sciences 63, 512–530 (2001)
Jiang, T., Li, M.: On the approximation of shortest common supersequence and longest common subsequences. SIAM Journal on Computing 24, 1112–1139 (1995)
Li, M., Ma, B., Wang, L.: On the closest string and substring problems. Journal of the ACM 49, 157–171 (2002)
Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal Of Computer and System Sciences 43, 425–440 (1991)
Papadimitriou, C., Yannakakis, M.: On limited nondeterminism and the complexity of VC dimension. Journal Of Computer and System Sciences 53, 161–170 (1996)
Papadimitriou, C., Yannakakis, M.: On the complexity of database queries. Journal Of Computer and System Sciences 58, 407–427 (1999)
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)
Sze, S.-H.: Lectures notes of Special Topics in Computational Biology, Texas A & M University (2002)
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Huang, X. (2006). Lower Bounds and Parameterized Approach for Longest Common Subsequence. In: Chen, D.Z., Lee, D.T. (eds) Computing and Combinatorics. COCOON 2006. Lecture Notes in Computer Science, vol 4112. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11809678_16
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DOI: https://doi.org/10.1007/11809678_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-36925-7
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