Abstract
In this paper, we compute the first linear kernel for the complementary problem of Maximal Strip Recovery (CMSR) — a well-known NP-complete problem in computational genomics. Let k be the parameter which represents the size of the solution. The core of the technique is to first obtain a tight 18k bound on the parameterized solution search space, which is done through a mixed global rules and local rules, and via an inverse amortized analysis. Then we apply additional data-reduction rules to obtain a tight 84k kernel for the problem. Combined with the known algorithm using bounded degree search, we obtain the best FPT algorithm for CMSR to this date, running in O(2.36k k 2 + n 2) time.
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
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On Problems without Polynomial Kernels (Extended Abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 563–574. Springer, Heidelberg (2008)
Bulteau, L., Fertin, G., Rusu, I.: Maximal Strip Recovery Problem with Gaps: Hardness and Approximation Algorithms. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 710–719. Springer, Heidelberg (2009)
Bulteau, L., Fertin, G., Jiang, M., Rusu, I.: Tractability and Approximability of Maximal Strip Recovery. In: Giancarlo, R., Manzini, G. (eds.) CPM 2011. LNCS, vol. 6661, pp. 336–349. Springer, Heidelberg (2011)
Chen, Z., Fu, B., Jiang, M., Zhu, B.: On recovering syntenic blocks from comparative maps. Journal of Combinatorial Optimization 18(3), 307–318 (2009)
Choi, V., Zheng, C., Zhu, Q., Sankoff, D.: Algorithms for the Extraction of Synteny Blocks from Comparative Maps. In: Giancarlo, R., Hannenhalli, S. (eds.) WABI 2007. LNCS (LNBI), vol. 4645, pp. 277–288. Springer, Heidelberg (2007)
Cook, S.: The complexity of theorem-proving procedures. In: Proceedings of the 3rd ACM Symp. on Theory of Computing (STOC 1971), pp. 151–158 (1971)
Downey, R., Fellows, M.: Parameterized Complexity. Springer (1999)
Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Proc. 42nd ACM Symp. Theory of Computation (STOC 2010), Cambridge, MA, USA, pp. 251–260 (2010)
Fellows, M.: The Lost Continent of Polynomial Time: Preprocessing and Kernelization. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 276–277. Springer, Heidelberg (2006)
Fernau, H., Fomin, F., Lokshtanov, D., Raible, D., Saurabh, S., Villanger, Y.: Kernel(s) for problems with no kernel: on out-trees with many leaves. In: Proc. 26th Intl. Symp. on Theoretical Aspects of Computer Science (STACS 2009), pp. 421–432 (2009)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)
Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: Proc. 40th ACM Symp. Theory of Computation (STOC 2008), Victoria, Canada, pp. 133–142 (2008)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38, 31–45 (2007)
Jiang, M.: Inapproximability of Maximal Strip Recovery. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 616–625. Springer, Heidelberg (2009)
Jiang, M.: Inapproximability of Maximal Strip Recovery: II. In: Lee, D.-T., Chen, D.Z., Ying, S. (eds.) FAW 2010. LNCS, vol. 6213, pp. 53–64. Springer, Heidelberg (2010)
Jiang, H., Li, Z., Lin, G., Wang, L., Zhu, B.: Exact and approximation algorithms for the complementary maximal strip recovery problem. J. of Combinatorial Optimization 23(4), 493–506 (2012)
Jiang, H., Zhang, C., Zhu, B.: Weak Kernels. ECCC Report, TR10-005 (October 2010)
Jiang, H., Zhu, B., Zhu, D.: Algorithms for sorting unsigned linear genomes by the DCJ operations. Bioinformatics 27, 311–316 (2011)
Li, Z., Goebel, R., Wang, L., Lin, G.: An Improved Approximation Algorithm for the Complementary Maximal Strip Recovery Problem. In: Atallah, M., Li, X.-Y., Zhu, B. (eds.) FAW-AAIM 2011. LNCS, vol. 6681, pp. 46–57. Springer, Heidelberg (2011)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Univ. Press (2006)
Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, NY (1972)
Wang, L., Zhu, B.: On the tractability of maximal strip recovery. J. of Computational Biology 17(7), 907–914 (2010); Correction 18(1), 129 (2011)
Zheng, C., Zhu, Q., Sankoff, D.: Removing noise and ambiguities from comparative maps in rearrangement analysis. IEEE/ACM Transactions on Computational Biology and Bioinformatics 4, 515–522 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jiang, H., Zhu, B. (2012). A Linear Kernel for the Complementary Maximal Strip Recovery Problem. In: Kärkkäinen, J., Stoye, J. (eds) Combinatorial Pattern Matching. CPM 2012. Lecture Notes in Computer Science, vol 7354. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31265-6_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-31265-6_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31264-9
Online ISBN: 978-3-642-31265-6
eBook Packages: Computer ScienceComputer Science (R0)