Abstract
Maximum stacking base pairs is a fundamental combinatorial problem from RNA secondary structure prediction under the energy model. The basic maximum stacking base pairs problem can be described as: given an RNA sequence, find a maximum number of base pairs such that each chosen base pair has at least one parallel and adjacent partner (i.e., they form a stacking). This problem is NP-hard, no matter whether the candidate base pairs follow the biology principle or are given explicitly as input. This paper investigates a restricted version of this problem where the base pairs are given as input and each base is associated with at most k (a constant) base pairs. We show that this restricted version is still APX-hard, even if the base pairs are weighted. Moreover, by a nonlinear LP-rounding method, we present an approximation algorithm with a factor \(\frac{32(k-1)^{3}e^{3}}{8(k-1)e-1}\). Applying our algorithms on the simulated data, the actual approximation factor is in fact much better than this theoretical bound.
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
References
Tinoco Jr., I., Bustamante, C.: How RNA folds. J. Mol. Biol. 293, 271–281 (1999)
Nussinov, R., Pieczenik, G., Griggs, J.R., Kleitman, D.J.: Algorithms for loop matchings. SIAM J. Appl. Math. 35(1), 68–82 (1978)
Nussinov, R., Jacobson, A.B.: Fast algorithm for predicting the secondary structure of single-stranded RNA. Proc. Natl. Acad. Sci. USA 77, 6309–6313 (1980)
Zuker, M., Stiegler, P.: Optimal computer folding of large RNA sequences using thermodynamics and auxiliary information. Nucleic Acids Res. 9, 133–148 (1981)
Zuker, M., Sankoff, D.: RNA secondary structures and their prediction. Bull. Math. Biol. 46, 591–621 (1984)
Sankoff, D.: Simultaneous solution of the RNA folding, alignment and protosequence problems. SIAM J. Appl. Math. 45, 810–825 (1985)
Lyngsø, R.B., Zuker, M., Pedersen, C.N.S.: Fast evaluation of interval loops in RNA secondary structure prediction. Bioinformatics 15, 440–445 (1999)
Lyngsø, R.B., Pedersen, C.N.S.: RNA pseudoknot prediction in energy based models. J. Comput. Biol. 7(3/4), 409–427 (2000)
Akutsu, T.: Dynamic programming algorithms for RNA secondary structure prediction with pseudoknots. Discrete Appl. Math. 104(1–3), 45–62 (2000)
Rivas, E., Eddy, S.R.: A dynamic programming algorithm for RNA structure prediction including pseudoknots. J. Mol. Biol. 285(5), 2053–2068 (1999)
Uemura, Y., Hasegawa, A., Kobayashi, S., Yokomori, T.: Tree adjoining grammars for RNA structure prediction. Theor. Comput. Sci. 210(2), 277–303 (1999)
Tinoco Jr., I., et al.: Improved estimation of secondary structure in ribonucleic acids. Nat. New Biol. 246, 40–42 (1973)
Ieong, S., Kao, M.-Y., Lam, T.-W., Sung, W.-K., Yiu, S.-M.: Predicting RNA secondary structure with arbitrary pseudoknots by maximizing the number of stacking pairs. J. Comput. Biol. 10, 981–995 (2003)
Lyngsø, R.B.: Complexity of pseudoknot prediction in simple models. In: DÃaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 919–931. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-27836-8_77
Jiang, M.: Approximation algorithms for predicting RNA secondary structures with arbitrary pseudoknots. IEEE/ACM Trans. Comput. Biol. Bioinform. 7(2), 323–332 (2010)
Berman, P.: A \(d\)/2 approximation for maximum weight independent set in \(d\)-Claw free graphs. Nordic J. Comput. 7, 178–184 (2000)
Zhou, A., Jiang, H., Guo, J., Zhu, D.: A new approximation algorithm for the maximum stacking base pairs problem from RNA secondary structures prediction. In: Gao, X., Du, H., Han, M. (eds.) COCOA 2017. LNCS, vol. 10627, pp. 85–92. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-71150-8_7
Berman, P., Karpinski, M.: On some tighter inapproximability results (extended abstract). In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 200–209. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48523-6_17
Acknowledgments
This research is supported by NSF of China under grant 61872427, 61732009 and 61628207, by NSF of Shandong Provence under grant ZR201702190130. Haitao Jiang is also supported by Young Scholars Program of Shandong University. Peiqiang Liu is also supported by Key Research and Development Program of Yantai City (2017ZH065) and CERNET Innovation Project (No. NGII20161204).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Liu, L., Jiang, H., Liu, P., Zhu, B., Zhu, D. (2019). Maximum Stacking Base Pairs: Hardness and Approximation by Nonlinear LP-Rounding. In: Cai, Z., Skums, P., Li, M. (eds) Bioinformatics Research and Applications. ISBRA 2019. Lecture Notes in Computer Science(), vol 11490. Springer, Cham. https://doi.org/10.1007/978-3-030-20242-2_21
Download citation
DOI: https://doi.org/10.1007/978-3-030-20242-2_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-20241-5
Online ISBN: 978-3-030-20242-2
eBook Packages: Computer ScienceComputer Science (R0)