A Unified Integer Programming Model for Genome Rearrangement Problems

  • Giuseppe Lancia
  • Franca Rinaldi
  • Paolo Serafini
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9043)

Abstract

We describe an integer programming (IP) model that can be applied to the solution of all genome-rearrangement problems in the literature. No direct IP model for such problems had ever been proposed prior to this work. Our model employs an exponential number of variables, but it can be solved by column generation techniques. I.e., we start with a small number of variables and we show how the correct missing variables can be added to the model in polynomial time.

Keywords

Genome rearrangements Evolutionary distance Sorting by reversals Sorting by transpositions Pancake flipping problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bader, M., Ohlebusch, E.: Sorting by weighted reversals, transpositions, and inverted transpositions. J. Comput. Biol. 14, 615–636 (2007)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bafna, V., Pevzner, P.: Genome rearrangements and sorting by reversals. SIAM J. Comp. 25, 272–289 (1996)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bafna, V., Pevzner, P.: Sorting by transpositions. SIAM J. Discr. Math. 11, 224–240 (1998)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Barnhart, C., Johnson, E.L., Nemhauser, G.L., Savelsbergh, M.W., Vance, P.H.: Branch-and-Price: Column Generation for Solving Huge Integer Programs. Op. Res. 46, 316–329 (1998)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Berman, P., Hannenhalli, S., Karpinski, M.: 1.375-Approximation algorithm for sorting by reversals. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 200–210. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Bulteau, L., Fertin, G., Rusu, I.: Pancake Flipping is Hard. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 247–258. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Bulteau, L., Fertin, G., Rusu, I.: Sorting by Transpositions Is Difficult. SIAM J. Discr. Math. 26, 1148Google Scholar
  8. 8.
    Caprara, A.: Sorting by reversals is difficult. In: 1st ACM/IEEE International Conference on Computational Molecular Biology, pp. 75–83. ACM Press (1997)Google Scholar
  9. 9.
    Caprara, A.: Sorting Permutations by Reversals and Eulerian Cycle Decompositions. SIAM J. on Disc. Math. 12, 91–110 (1999)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Caprara, A., Lancia, G., Ng, S.-K.: A Column-Generation Based Branch-and-Bound Algorithm for Sorting By Reversals. In: Mathematical Support For Molecular Biology. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 47, pp. 213–226 (1999)Google Scholar
  11. 11.
    Caprara, L.G., Ng, S.K.: Sorting Permutations by Reversals through Branch and Price. INFORMS J. on Comp. 13, 224–244 (2001)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Chitturi, B., Fahle, W., Meng, Z., Morales, L., Shields, C.O., Sudborough, I.H., Voit, W.: An 18/11 n upper bound for sorting by prefix reversals. Theor. Comp. Sc. 410, 3372–3390 (2009)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Christie, A.: A 3/2-approximation algorithm for sorting by reversals. In: 9th ACM-SIAM Symposium on Discrete Algorithms, pp. 244–252. ACM Press (1998)Google Scholar
  14. 14.
    Fischer, J., Ginzinger, S.W.: A 2-Approximation Algorithm for Sorting by Prefix Reversals. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 415–425. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Gates, W., Papadimitriou, C.: Bounds for sorting by prefix reversal. Discr. Math. 27, 47–57 (1979)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Gu, Q.P., Peng, S., Sudborough, H.: A 2-approximation algorithm for genome rearrangements by reversals and transpositions. Theoret. Comput. Sci. 210, 327–339 (1999)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Hartman, T., Sharan, R.: A 1.5-approximation algorithm for sorting by transpositions and transreversals. J. Comput. Syst. Sci. 70, 300–320 (2005)Google Scholar
  18. 18.
    Kececioglu, J., Sankoff, D.: Exact and approximation algorithms for sorting by reversals, with application to genome rearrangement. Algorithmica 13, 180–210 (1995)Google Scholar
  19. 19.
    Lancia, G., Serafini, P.: Deriving compact extended formulations via LP-based separation techniques. 4OR 12, 201–234 (2014)Google Scholar
  20. 20.
    Meidanis, J., Walter, M.M.T., Dias, Z.: A Lower Bound on the Reversal and Transposition Diameter. J. Comput. Biol. 9, 743–745 (2002)CrossRefGoogle Scholar
  21. 21.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization, 784 pages. Wiley (1999)Google Scholar
  22. 22.
    Sankoff, D., Cedergren, R., Abel, Y.: Genomic divergence through gene rearrangement. In: Molecular Evolution: Computer Analysis of Protein and Nucleic Acid Sequences, pp. 428–438. Academic Press, New York (1990)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Giuseppe Lancia
    • 1
  • Franca Rinaldi
    • 1
  • Paolo Serafini
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversity of UdineUdineItaly

Personalised recommendations