Advertisement

On the Fixed Parameter Tractability and Approximability of the Minimum Error Correction Problem

  • Paola Bonizzoni
  • Riccardo Dondi
  • Gunnar W. Klau
  • Yuri Pirola
  • Nadia Pisanti
  • Simone ZaccariaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9133)

Abstract

Haplotype assembly is the computational problem of reconstructing the two parental copies, called haplotypes, of each chromosome starting from sequencing reads, called fragments, possibly affected by sequencing errors. Minimum Error Correction (MEC) is a prominent computational problem for haplotype assembly and, given a set of fragments, aims at reconstructing the two haplotypes by applying the minimum number of base corrections.

By using novel combinatorial properties of MEC instances, we are able to provide new results on the fixed-parameter tractability and approximability of MEC. In particular, we show that MEC is in FPT when parameterized by the number of corrections, and, on “gapless” instances, it is in FPT also when parameterized by the length of the fragments, whereas the result known in literature forces the reconstruction of complementary haplotypes. Then, we show that MEC cannot be approximated within any constant factor while it is approximable within factor \(O(\log nm)\) where \(n m\) is the size of the input. Finally, we provide a practical 2-approximation algorithm for the Binary MEC, a variant of MEC that has been applied in the framework of clustering binary data.

Notes

Acknowledgements

This work has been stimulated by discussions between PB, GK, and NP during the No.045 NII Shonan workshop on Exact Algorithms for Bioinformatics Research, March 2014, Japan.

The authors acknowledge the support of the MIUR PRIN 2010-2011 grant 2010LYA9RH (Automi e Linguaggi Formali: Aspetti Matematici e Applicativi), of the Cariplo Foundation grant 2013-0955 (Modulation of anti cancer immune response by regulatory non-coding RNAs), of the FA 2013 grant (Metodi algoritmici e modelli: aspetti teorici e applicazioni in bioinformatica).

References

  1. 1.
    Aguiar, D., Istrail, S.: HapCompass: a fast cycle basis algorithm for accurate haplotype assembly of sequence data. J. Comput. Biol. 19(6), 577–590 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bansal, V., Bafna, V.: HapCUT: an efficient and accurate algorithm for the haplotype assembly problem. Bioinformatics 24(16), i153–i159 (2008)CrossRefGoogle Scholar
  3. 3.
    Bonizzoni, P., Della Vedova, G., Dondi, R., Li, J.: The haplotyping problem: an overview of computational models and solutions. J. Comput. Sci. Techol. 18(6), 675–688 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Browning, B., Browning, S.: Haplotypic analysis of Wellcome Trust case control consortium data. Hum. Genet. 123(3), 273–280 (2008)CrossRefGoogle Scholar
  5. 5.
    Chen, Z.Z., Deng, F., Wang, L.: Exact algorithms for haplotype assembly from whole-genome sequence data. Bioinformatics 29(16), 1938–45 (2013)CrossRefGoogle Scholar
  6. 6.
    Cilibrasi, R., Van Iersel, L., Kelk, S., Tromp, J.: The complexity of the single individual SNP haplotyping problem. Algorithmica 49(1), 13–36 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Dondi, R.: New results for the Longest Haplotype Reconstruction problem. Discrete Appl. Math. 160(9), 1299–1310 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Fouilhoux, P., Mahjoub, A.: Solving VLSI design and DNA sequencing problems using bipartization of graphs. Comput. Optim. Appl. 51(2), 749–781 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  10. 10.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Approximate max-flow min-(multi) cut theorems and their applications. SIAM J. Comput. 25(2), 235–251 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Guo, J., et al.: Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J. Comput. Syst. Sci. 72(8), 1386–1396 (2006)zbMATHCrossRefGoogle Scholar
  12. 12.
    Halldórsson, B.V., Aguiar, D., Istrail, S.: Haplotype phasing by multi-assembly of shared haplotypes: phase-dependent interactions between rare variants. In: PSB, pp. 88–99. World Scientific Publishing (2011)Google Scholar
  13. 13.
    He, D., et al.: Optimal algorithms for haplotype assembly from whole-genome sequence data. Bioinformatics 26(12), i183–i190 (2010)CrossRefGoogle Scholar
  14. 14.
    Jiao, Y., Xu, J., Li, M.: On the k-closest substring and k-consensus pattern problems. In: Sahinalp, S.C., Muthukrishnan, S.M., Dogrusoz, U. (eds.) CPM 2004. LNCS, vol. 3109, pp. 130–144. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  15. 15.
    Khot, S.: On the power of unique 2-prover 1-round games. In: STOC, pp. 767–775. ACM (2002)Google Scholar
  16. 16.
    Kleinberg, J., Papadimitriou, C., Raghavan, P.: Segmentation problems. In: STOC, pp. 473–482. ACM (1998)Google Scholar
  17. 17.
    Lancia, G., Bafna, V., Istrail, S., Lippert, R., Schwartz, R.: SNPs problems, complexity, and algorithms. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 182–193. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  18. 18.
    Lippert, R., Schwartz, R., Lancia, G., Istrail, S.: Algorithmic strategies for the single nucleotide polymorphism haplotype assembly problem. Brief. Bioinform. 3(1), 23–31 (2002)CrossRefGoogle Scholar
  19. 19.
    Ostrovsky, R., Rabani, Y.: Polynomial-time approximation schemes for geometric min-sum median clustering. J. ACM 49(2), 139–156 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Patterson, M., Marschall, T., Pisanti, N., van Iersel, L., Stougie, L., Klau, G.W., Schönhuth, A.: WhatsHap: haplotype assembly for future-generation sequencing reads. In: Sharan, R. (ed.) RECOMB 2014. LNCS (LNBI), vol. 8394, pp. 237–249. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  21. 21.
    Pirola, Y., Bonizzoni, P., Jiang, T.: An efficient algorithm for haplotype inference on pedigrees with recombinations and mutations. IEEE/ACM Trans. Comput. Biol. Bioinform. 9(1), 12–25 (2012)CrossRefGoogle Scholar
  22. 22.
    Pirola, Y., et al.: Haplotype-based prediction of gene alleles using pedigrees and SNP genotypes. In: BCB, pp. 33–41. ACM (2013)Google Scholar
  23. 23.
    Reed, B., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Paola Bonizzoni
    • 1
  • Riccardo Dondi
    • 2
  • Gunnar W. Klau
    • 3
    • 5
  • Yuri Pirola
    • 1
  • Nadia Pisanti
    • 4
    • 5
  • Simone Zaccaria
    • 1
    Email author
  1. 1.DISCoUniv. degli Studi di Milano-BicoccaMilanItaly
  2. 2.Dip. di Scienze Umane e SocialiUniv. degli Studi di BergamoBergamoItaly
  3. 3.Life Sciences, Centrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands
  4. 4.Dipartimento di InformaticaUniv. degli Studi di PisaPisaItaly
  5. 5.Erable Team, INRIALyonFrance

Personalised recommendations