Advertisement

On the Number of Solutions of the Discretizable Molecular Distance Geometry Problem

  • Leo Liberti
  • Benoît Masson
  • Jon Lee
  • Carlile Lavor
  • Antonio Mucherino
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6831)

Abstract

The Discretizable Molecular Distance Geometry Problem is a subset of instances of the distance geometry problem that can be solved by a combinatorial algorithm called “Branch-and-Prune”. It was observed empirically that the number of solutions of YES instances is always a power of two. We perform an extensive theoretical analysis of the number of solutions for these instances and we prove that this number is a power of two with probability one.

Keywords

distance geometry symmetry Branch-and-Prune power of two 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lavor, C., Liberti, L., Maculan, N.: Computational experience with the molecular distance geometry problem. In: Pintér, J. (ed.) Global Optimization: Scientific and Engineering Case Studies, pp. 213–225. Springer, Berlin (2006)CrossRefGoogle Scholar
  2. 2.
    Liberti, L., Lavor, C., Maculan, N., Marinelli, F.: Double variable neighbourhood search with smoothing for the molecular distance geometry problem. Journal of Global Optimization 43, 207–218 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Saxe, J.: Embeddability of weighted graphs in k-space is strongly NP-hard. In: Proceedings of 17th Allerton Conference in Communications, Control and Computing, pp. 480–489 (1979)Google Scholar
  4. 4.
    Huang, H.X., Liang, Z.A., Pardalos, P.: Some properties for the Euclidean distance matrix and positive semidefinite matrix completion problems. Journal of Global Optimization 25, 3–21 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hendrickson, B.: The molecule problem: exploiting structure in global optimization. SIAM Journal on Optimization 5, 835–857 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eren, T., Goldenberg, D., Whiteley, W., Yang, Y., Morse, A., Anderson, B., Belhumeur, P.: Rigidity, computation, and randomization in network localization. IEEE Infocom Proceedings, 2673–2684 (2004)Google Scholar
  7. 7.
    Krislock, N., Wolkowicz, H.: Explicit sensor network localization using semidefinite representations and facial reductions. SIAM Journal on Optimization 20, 2679–2708 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gunther, H.: NMR Spectroscopy: Basic Principles, Concepts, and Applications in Chemistry. Wiley, New York (1995)Google Scholar
  9. 9.
    Schlick, T.: Molecular modelling and simulation: an interdisciplinary guide. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  10. 10.
    Santana, R., Larrañaga, P., Lozano, J.: Combining variable neighbourhood search and estimation of distribution algorithms in the protein side chain placement problem. Journal of Heuristics 14, 519–547 (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Lavor, C., Mucherino, A., Liberti, L., Maculan, N.: Discrete approaches for solving molecular distance geometry problems using NMR data. International Journal of Computational Biosciences 1(1), 88–94 (2010)Google Scholar
  12. 12.
    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: The discretizable molecular distance geometry problem. Computational Optimization and Applications doi: 10.1007/s10589-011-9402-6Google Scholar
  13. 13.
    Liberti, L., Lavor, C., Maculan, N.: A branch-and-prune algorithm for the molecular distance geometry problem. International Transactions in Operational Research 15, 1–17 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mucherino, A., Lavor, C., Liberti, L.: The discretizable distance geometry problem. To appear in Optimization LettersGoogle Scholar
  15. 15.
    Lavor, C., Lee, J., John, A.L.S., Liberti, L., Mucherino, A., Sviridenko, M.: Discretization orders for distance geometry problems. Optimization Letters doi: 10.1007/s11590-011-0302-6Google Scholar
  16. 16.
    Lavor, C., Mucherino, A., Liberti, L., Maculan, N.: On the computation of protein backbones by using artificial backbones of hydrogens. Journal of Global Optimization 50, 329–344 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liberti, L., Lavor, C., Mucherino, A., Maculan, N.: Molecular distance geometry methods: from continuous to discrete. International Transactions in Operational Research 18, 33–51 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: Recent advances on the discretizable molecular distance geometry problem. European Journal of Operational Research (accepted / invited survey)Google Scholar
  19. 19.
    Blumenthal, L.: Theory and Applications of Distance Geometry. Oxford University Press, Oxford (1953)zbMATHGoogle Scholar
  20. 20.
    Connelly, R.: Generic global rigidity. Discrete Computational Geometry 33, 549–563 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Brady, T., Watt, C.: On products of Euclidean reflections. American Mathematical Monthly 113, 826–829 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lavor, C., Liberti, L., Maculan, N.: The discretizable molecular distance geometry problem. Technical Report q-bio/0608012, arXiv (2006)Google Scholar
  23. 23.
    Dong, Q., Wu, Z.: A geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance data. Journal of Global Optimization 26, 321–333 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Coope, I.: Reliable computation of the points of intersection of n spheres in ℝn. Australian and New Zealand Industrial and Applied Mathematics Journal 42, C461–C477 (2000)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Leo Liberti
    • 1
  • Benoît Masson
    • 2
  • Jon Lee
    • 3
  • Carlile Lavor
    • 4
  • Antonio Mucherino
    • 5
  1. 1.LIX, École PolytechniquePalaiseauFrance
  2. 2.IRISA, INRIA, Campus de BeaulieuRennesFrance
  3. 3.Dept. of Mathematical SciencesIBM T.J. Watson Research CenterYorktown HeightsUSA
  4. 4.Department of Applied Mathematics (IMECC-UNICAMP)State University of CampinasCampinasBrazil
  5. 5.CERFACSToulouseFrance

Personalised recommendations