Journal of Combinatorial Optimization

, Volume 16, Issue 2, pp 127–154 | Cite as

A Lagrangian relaxation approach for the multiple sequence alignment problem

  • Ernst Althaus
  • Stefan CanzarEmail author
Open Access


We present a branch-and-bound (bb) algorithm for the multiple sequence alignment problem (MSA), one of the most important problems in computational biology. The upper bound at each bb node is based on a Lagrangian relaxation of an integer linear programming formulation for MSA. Dualizing certain inequalities, the Lagrangian subproblem becomes a pairwise alignment problem, which can be solved efficiently by a dynamic programming approach. Due to a reformulation w.r.t. additionally introduced variables prior to relaxation we improve the convergence rate dramatically while at the same time being able to solve the Lagrangian problem efficiently. Our experiments show that our implementation, although preliminary, outperforms all exact algorithms for the multiple sequence alignment problem. Furthermore, the quality of the alignments is among the best computed so far.


Sequence comparison Lagrangian relaxation Branch and bound 


  1. Althaus E, Caprara A, Lenhof H-P, Reinert K (2002) Multiple sequence alignment with arbitrary gap costs: Computing an optimal solution using polyhedral combinatorics. In: Lengauer T, Lenhof H-P (eds) Proceedings of the European conference on computational biology, Saarbrücken, October 2002. Bioinformatics, vol 18. Oxford University Press, London, pp S4–S16 Google Scholar
  2. Althaus E, Caprara A, Lenhof H-P, Reinert K (2006) A branch-and-cut algorithm for multiple sequence alignment. Math Program 105:387–425 zbMATHCrossRefMathSciNetGoogle Scholar
  3. Beasley J (1993) Lagrangian relaxation. In: Modern heuristic techniques for combinatorial problems. Blackwell Scientific, Oxford Google Scholar
  4. Caprara A, Fischetti M, Toth P (1999) A heuristic method for the set cover problem. Oper Res 47:730–743 zbMATHMathSciNetGoogle Scholar
  5. Carrillo H, Lipman DJ (1988) The multiple sequence alignment problem in biology. SIAM J Appl Math 48(5):1073–1082 zbMATHCrossRefMathSciNetGoogle Scholar
  6. Delcher A, Kasif S, Fleischmann R, Peterson J, White O, Salzberg S (1999) Alignment of whole genomes. Nucleic Acids Res 27:2369–2376 CrossRefGoogle Scholar
  7. Edgar RC (2004) Muscle: multiple sequence alignment with high accuracy and high throughput. Nucleic Acids Res 32(5):1792–1797 CrossRefGoogle Scholar
  8. Elias I (2003) Settling the intractability of multiple alignment. In: Proc. of the 14th ann. int. symp. on algorithms and computation (ISAAC’03). Lecture notes in computer science, vol 2906. Springer, Berlin, pp 352–363 Google Scholar
  9. Eppstein D (1990) Sequence comparison with mixed convex and concave costs. J Algorithms 11:85–101 zbMATHCrossRefMathSciNetGoogle Scholar
  10. Fisher M (1994) Optimal solutions of vehicle routing problems using minimum k-trees. Oper Res 42:626–642 zbMATHMathSciNetCrossRefGoogle Scholar
  11. Garey M, Johnson D (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, New York zbMATHGoogle Scholar
  12. Gupta S, Kececioglu J, Schaeffer A (1995) Improving the practical space and time efficiency of the shortest-paths approach to sum-of-pairs multiple sequence alignment. J Comput Biol 2:459–472 Google Scholar
  13. Gusfield D (1997) Algorithms on strings, trees and sequences: computer science and computational biology. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  14. Held M, Karp R (1971) The traveling salesman problem and minimum spanning trees: part II. Math Program 1:6–25 zbMATHCrossRefMathSciNetGoogle Scholar
  15. Katoh K, Kuma K, Toh H, Miyata T (2005) MAFFT version 5: improvement in accuracy of multiple sequence alignment. Nucleic Acids 33:511 CrossRefGoogle Scholar
  16. Larmore L, Schieber B (1990) Online dynamic programming with applications to the prediction of RNA secondary structure. In: Proceedings of the first symposium on discrete algorithms, pp 503–512 Google Scholar
  17. Lermen M, Reinert K (2000) The practical use of the \(\mathcal{A}^{*}\) algorithm for exact multiple sequence alignment. J Comput Biol 7(5):655–673 CrossRefGoogle Scholar
  18. Lipman D, Altschul S, Kececioglu J (1989) A tool for multiple sequence alignment. Proc Nat Acad Sci US Am 86:4412–4415 CrossRefGoogle Scholar
  19. Lucena A (1993) Steiner problem in graphs: Lagrangean relaxation and cutting-planes. COAL Bull 21:2–7 Google Scholar
  20. Mehlhorn K, Näher S (1999) The LEDA platform of combinatorial and geometric computing. Cambridge University Press, Cambridge. See also Google Scholar
  21. Notredame C, Higgins DG, Heringa J (2000) T-Coffee: a novel method for fast and accurate multiple sequence alignment. J Mol Biol 302(1):205–217 CrossRefGoogle Scholar
  22. Reinert K (1999) A polyhedral approach to sequence alignment problems. PhD thesis, Universität des Saarlandes, 1999 Google Scholar
  23. Reinert K, Lenhof H-P, Mutzel P, Mehlhorn K, Kececioglu J (1997) A branch-and-cut algorithm for multiple sequence alignment. In: Proceedings of the first annual international conference on computational molecular biology (RECOMB-97), pp 241–249 Google Scholar
  24. Reinert K, Stoye J, Will T (2000) An iterative method for faster sum-of-pairs multiple sequence alignment. Bioinformatics 16(9):808–814 CrossRefGoogle Scholar
  25. Sankoff D, Kruskal JB (1983) Time warps, string edits and macromolecules: the theory and practice of sequence comparison. Addison–Wesley, Reading Google Scholar
  26. Subramanian AR, Weyer-Menkhoff J, Kaufmann M, Morgenstern B (2005) DIALIGN-T: An improved algorithm for segment-based multiple sequence alignment. BMC Bioinformatics 6:66 CrossRefGoogle Scholar
  27. Thompson JD, Higgins DG, Gibson TJ (1994) CLUSTAL W: improving the sensitivity of progressive multiple sequence alignment through sequence weighting, position-specific gap penalties and weight matrix choice. Nucleic Acids Res 22(22):4673–4680 CrossRefGoogle Scholar

Copyright information

© The Author(s) 2008

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Max-Planck Institut für InformatikSaarbrückenGermany

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