Abstract
The Closest Substring problem (CSP), where a short string is sought that minimizes the number of mismatches between it and each of a given set of strings, is a minimization problem with polynomial time approximation schemes. In this paper, a lower bound on approximation algorithms for the CSP problem is developed. We prove that unless the Exponential Time Hypothesis (ETH Hypothesis, i.e., not all search problems in SNP are solvable in subexponential time) fails, the CSP problem has no polynomial time approximation schemes of running time f(1/ε)|x|O(1/ ∈ ) for any function f, where |x| is the size of input instance. This essentially excludes the possibility that the CSP problem has a practical polynomial time approximation scheme even for moderate values of the error bound ε . As a consequence, it is unlikely that the study of approximation schemes for the CSP problem in the literature would lead to practical approximation algorithms for the CSP problem for small error bound ε.
This work is supported by the National Natural Science Foundation of China (60433020), the Program for New Century Excellent Talents in University (NCET-05-0683) and the Program for Changjiang Scholars and Innovative Research Team in University (IRT0661).
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References
Lanctot, J.K., Li, M., Ma, B., Wang, S., Zhang, L.: Distinguishing string selection problems. Information and Computation 185(1), 41–55 (2003)
Li, M., Ma, B., Wang, L.: On the closest string and substring problems. Journal of the ACM 49(2), 157–171 (2002)
Ma, B.: A polynomial time approximation scheme for the closest substring problem. In: Giancarlo, R., Sankoff, D. (eds.) CPM 2000. LNCS, vol. 1848, pp. 99–107. Springer, Heidelberg (2000)
Downey, R., Fellows, M.: Parameterized Complexity. Springer, New York (1999)
Chen, J.: Parameterized computation and complexity: a new approach dealing with NP-hardness. Journal of Computer Science and Technology 20(1), 18–37 (2005)
Chen, J., Huang, X., Kanj, I., Xia, G.: Linear FPT reductions and computational lower bounds. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC 2004), pp. 212–221. ACM Press, New York (2004)
Chen, J., Huang, X., Kanj, I., Xia, G.: W-hardness under linear FPT-reductions: structural properties and further applications. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 16–19. Springer, Heidelberg (2005)
Fellows, M., Gramm, J., Niedermeier, R.: On the parameterized intractability of Closest Substring and related problems. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 262–273. Springer, Heidelberg (2002)
Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. In: Proceedings of the 12th Annual ACM Symposium on Theory of Computing (STOC 1988), pp. 229–234. ACM Press, New York (1988)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? Journal of Computer and System Sciences 63(4), 512–530 (2001)
Garey, M., Johnson, D.: Computers and Intractability. A Guide to the Theory of NP-Completness. W.H. Freeman and Company, New York (1979)
Gramm, J., Guo, J., Niedermeier, R.: On exact and approximation algorithms for distinguishing substring selection. In: Lingas, A., Nilsson, B.J. (eds.) FCT 2003. LNCS, vol. 2751, pp. 195–209. Springer, Heidelberg (2003)
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Wang, J., Huang, M., Chen, J. (2007). A Lower Bound on Approximation Algorithms for the Closest Substring Problem. In: Dress, A., Xu, Y., Zhu, B. (eds) Combinatorial Optimization and Applications. COCOA 2007. Lecture Notes in Computer Science, vol 4616. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73556-4_31
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DOI: https://doi.org/10.1007/978-3-540-73556-4_31
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