Abstract
The problem of finding the Shortest Common Supersequence (SCS) of an arbitrary number of input strings is a well-studied problem. Given a set L of k strings, s 1, s 2, ..., s k , over an alphabet Σ, we say that their SCS is the shortest string that contains each of the input strings as a subsequence. The problem is known to be NP-hard [8] even over binary alphabet [12]. In this paper we focus on approximating two NP-hard variants of the SCS problem. For the first variant, where all input strings are of length 2, we present a \(2 - \frac {2}{1 + \log{n}\log{\log{n}}}\) approximation algorithm, where |Σ| = n. This result immediately improves the \(2 - \frac {4}{n+1}\) approximation algorithm presented in [17]. Moreover, we present a \(\frac{7}{6}\) (\(\approx 1.166\bar{6}\)) approximation algorithm for the restricted variant (but still NP-hard) where all input strings are of length 2 and every character in Σ has at most 3 occurrences in L.
Keywords
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bafna, V., Lawler, E.L., Pevzner, P.A.: Approximation Algorithms for Multiple Sequence Alignment. Theoretical Computer Science 182(1-2), 233–244 (1997)
Barone, P., Bonizzoni, P., Vedova, G.D., Mauri, G.: An approximation algorithm for the shortest common supersequence problem: an experimental analysis. In: ACM Symposium on Applied Computing, pp. 56–60 (2001)
Cotta, C.: A Comparison of Evolutionary Approaches to the Shortest Common Supersequence Problem. In: Cabestany, J., Prieto, A.G., Sandoval, F. (eds.) IWANN 2005. LNCS, vol. 3512, pp. 50–58. Springer, Heidelberg (2005)
Cotta, C.: Memetic Algorithms with Partial Lamarckism for the Shortest Common Supersequence Problem. In: Mira, J., Álvarez, J.R. (eds.) IWINAC 2005. LNCS, vol. 3562, pp. 84–91. Springer, Heidelberg (2005)
Even, G., Naor, J., Schieber, B., Sudan, M.: Approximating Minimum Feedback Sets and Multicuts in Directed Graphs. Algorithmica 20(2), 151–174 (1998)
Fraser, C.B., Irving, R.W.: Approximation Algorithms for the Shortest Common Supersequence. Nordic Journal of Computing 2(3), 303–325 (1995)
Jiang, T., Li, M.: On the Approximation of Shortest Common Supersequences and Longest Common Subsequences. SIAM Journal on Computing 24(5), 1122–1139 (1995)
Maier, D.: The Complexity of Some Problems on Subsequences and Supersequences. Journal of the ACM 25(2), 322–336 (1978)
Middendorf, M.: The Shortest Common Nonsubsequence Problem is NP-Complete. Theoretical Computer Science 108(2), 365–369 (1993)
Pevzner, P.A.: Multiple Alignment, Communication Cost, and Graph Matching. SIAM Journal on Applied Mathematics 52(6), 1763–1779 (1992)
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(4), 757–771 (2003)
Räihä, K.J., Ukkonen, E.: The Shortest Common Supersequence Problem over Binary Alphabet is NP-Complete. Theoretical Computer Science 16, 187–198 (1981)
Reingold, E.M., Nievergelt, J., Deo, N.: Combinatorial Algorithms. Prentice-Hall Inc., Englewood Cliffs (1977)
Rubinov, A.R., Timkovsky, V.G.: String Noninclusion Optimization Problems. SIAM Journal on Discrete Mathematics 11(3), 456–467 (1998)
Sankoff, D.: Minimal Mutation Trees of Sequences. SIAM Journal on Applied Mathematics 28, 35–42 (1975)
Timkovsky, V.G.: Complexity of common subsequence and supersequence problems and related problems. Kibernetika 5, 1–13 (1989); English Translation in Cybernetics 25, 565–580 (1990)
Timkovsky, V.G.: Some Approximations for Shortest Common Nonsubsequences and Supersequences. In: Amir, A., Turpin, A., Moffat, A. (eds.) SPIRE 2008. LNCS, vol. 5280, pp. 258–269. Springer, Heidelberg (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gotthilf, Z., Lewenstein, M. (2009). Improved Approximation Results on the Shortest Common Supersequence Problem. In: Karlgren, J., Tarhio, J., Hyyrö, H. (eds) String Processing and Information Retrieval. SPIRE 2009. Lecture Notes in Computer Science, vol 5721. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03784-9_27
Download citation
DOI: https://doi.org/10.1007/978-3-642-03784-9_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03783-2
Online ISBN: 978-3-642-03784-9
eBook Packages: Computer ScienceComputer Science (R0)