Abstract
The Closest String problem asks to find a string s which is not too far from each string in a set of m input strings, where the distance is taken as the Hamming distance. This well-studied problem has various applications in computational biology and drug design. In this paper, we introduce a new variant of Closest String where the input strings can contain wildcards that can match any letter in the alphabet, and the goal is to find a solution string without wildcards. We call this problem the Closest String with Wildcards problem, and we analyze it in the framework of parameterized complexity. Our study determines for each natural parameterization whether this parameterization yields a fixed-parameter algorithm, or whether such an algorithm is highly unlikely to exist.
More specifically, let m denote the number of input strings, each of length n, and let d be the given distance bound for the solution string. Furthermore, let k denote the minimum number of wildcards in any input string. We present fixed-parameter algorithms for the parameters m, n, and k + d, respectively. On the other hand, we then show that such results are unlikely to exist when k and d are taken as single parameters. This is done by showing that the problem is NP-hard already for k = 0 and d ≥ 2. Finally, to complement the latter result, we present a polynomial-time algorithm for the case of d = 1. Apart from this last result, all other results hold even when the strings are over a general alphabet.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Amir, A., Paryenty, H., Roditty, L.: Approximations and partial solutions for the consensus sequence problem. In: Grossi, R., Sebastiani, F., Silvestri, F. (eds.) SPIRE 2011. LNCS, vol. 7024, pp. 168–173. Springer, Heidelberg (2011)
Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Information Processing Letters 8(3), 121–123 (1979)
Chen, Z.Z., Wang, L.: Fast exact algorithms for the closest string and substring problems with application to the planted (l, d)-motif model. IEEE/ACM Transactions on Computational Biology and Bioinformatics 8(5), 1400–1410 (2011)
Chen, Z.Z., Ma, B., Wang, L.: A three-string approach to the closest string problem. Journal of Computer and System Sciences 78(1), 164–178 (2012)
Chimani, M., Woste, M., Böcker, S.: A closer look at the closest string and closest substring problem. In: ALENEX, pp. 13–24 (2011)
Dopazo, J., Rodríguez, A., Sáiz, J.C., Sobrino, F.: Design of primers for pcr ampiification of highly variable genomes. Computer Applications in the Biosciences: CABIOS 9(2), 123–125 (1993)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)
Even, S., Itai, A., Shamir, A.: On the complexity of timetable and multicommodity flow problems. SIAM Journal of Computing 5(4), 691–703 (1976)
Fellows, M.R., Gramm, J., Niedermeier, R.: On the parameterized intractability of motif search problems*. Combinatorica 26(2), 141–167 (2006)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)
Frances, M., Litman, A.: On covering problems of codes. Theory of Computing Systems 30(2), 113–119 (1997)
Gramm, J., Hüffner, F., Niedermeie, R.: Closest strings, primer design, and motif search. In: Currents in Computational Molecular Biology, Poster Abstracts of RECOMB, vol. 2002, pp. 74–75 (2002)
Gramm, J., Niedermeier, R., Rossmanith, P.: Fixed-parameter algorithms for closest string and related problems. Algorithmica 37(1), 25–42 (2003)
Karp, R.M.: Reducibility among combinatorial problems. Complexity of Computer Computations, 85–103 (1972)
Lanctot, J.K., Li, M., Ma, B., Wang, S., Zhang, L.: Distinguishing string selection problems. In: Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 633–642 (1999)
Lenstra, W.: Integer programming with a fixed number of variables. Mathematics of Operations Research, 538–548 (1983)
Li, M., Ma, B., Wang, L.: On the closest string and substring problems. Journal of the ACM (JACM) 49(2), 157–171 (2002)
Ma, B., Sun, X.: More efficient algorithms for closest string and substring problems. SIAM Journal on Computing 39(4), 1432–1443 (2009)
Marx, D.: Closest substring problems with small distances. SIAM Journal on Computing 38(4), 1382–1410 (2008)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)
Proutski, V., Holmes, E.C.: Primer master: A new program for the design and analysis of pcr primers. Computer Applications in the Biosciences: CABIOS 12, 253–255 (1996)
Stojanovic, N., Berman, P., Gumucio, D., Hardison, R., Miller, W.: A linear-time algorithm for the 1-mismatch problem. In: Rau-Chaplin, A., Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 1997. LNCS, vol. 1272, pp. 126–135. Springer, Heidelberg (1997)
Wang, L., Zhu, B.: Efficient algorithms for the closest string and distinguishing string selection problems. In: Deng, X., Hopcroft, J.E., Xue, J. (eds.) FAW 2009. LNCS, vol. 5598, pp. 261–270. Springer, Heidelberg (2009)
Zhao, R., Zhang, N.: A more efficient closest string problem. In: Bioinformatics and Computational Biology, pp. 210–215 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Hermelin, D., Rozenberg, L. (2014). Parameterized Complexity Analysis for the Closest String with Wildcards Problem. In: Kulikov, A.S., Kuznetsov, S.O., Pevzner, P. (eds) Combinatorial Pattern Matching. CPM 2014. Lecture Notes in Computer Science, vol 8486. Springer, Cham. https://doi.org/10.1007/978-3-319-07566-2_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-07566-2_15
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07565-5
Online ISBN: 978-3-319-07566-2
eBook Packages: Computer ScienceComputer Science (R0)