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Optimal Discretization Orders for Distance Geometry: A Theoretical Standpoint

  • Antonio Mucherino
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9374)

Abstract

Distance geometry consists in embedding a simple weighted undirected graph \(G=(V,E,d)\) in a K-dimensional space so that all distances \(d_{uv}\), which are the weights on the edges of G, are satisfied by the positions assigned to its vertices. The search domain of this problem is generally continuous, but it can be discretized under certain assumptions, that are strongly related to the order given to the vertices of G. This paper formalizes the concept of optimal partial discretization order, and adapts a previously proposed algorithm with the aim of finding discretization orders that are also able to optimize a given set of objectives. The objectives are conceived for improving the structure of the discrete search domain, for its exploration to become more efficient.

Notes

Acknowledgments

I am thankful to Douglas S. Gonçalves and Leo Liberti for the fruitful discussions.

References

  1. 1.
    Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cassioli, A., Günlük, O., Lavor, C., Liberti, L.: Discretization vertex orders in distance geometry. Discrete Appl. Math. 197, 27–41 (2015). doi: 10.1016/j.dam.2014.08.035 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Costa, V., Mucherino, A., Lavor, C., Cassioli, A., Carvalho, L.M., Maculan, N.: Discretization orders for protein side chains. J. Global Optim. 60(2), 333–349 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gonçalves, D.S., Mucherino, A.: Discretization orders and efficient computation of cartesian coordinates for distance geometry. Optim. Lett. 8(7), 2111–2125 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cabalar, P.: Answer set; programming? In: Balduccini, M., Son, T.C. (eds.) Logic Programming, Knowledge Representation, and Nonmonotonic Reasoning. LNCS, vol. 6565, pp. 334–343. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  6. 6.
    Lavor, C., Lee, J., Lee-St.John, A., Liberti, L., Mucherino, A., Sviridenko, M.: Discretization orders for distance geometry problems. Optim. Lett. 6(4), 783–796 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52, 115–146 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lavor, C., Liberti, L., Mucherino, A.: The interval branch-and-prune algorithm for the discretizable molecular distance geometry problem with inexact distances. J. Global Optim. 56(3), 855–871 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liberti, L., Lavor, C., Maculan, N.: A branch-and-prune algorithm for the molecular distance geometry problem. Int. Trans. Oper. Res. 15, 1–17 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56(1), 3–69 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liberti, L., Lavor, C., Mucherino, A., Maculan, N.: Molecular distance geometry methods: from continuous to discrete. Int. Trans. Oper. Res. 18(1), 33–51 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mucherino, A.: On the identification of discretization orders for distance geometry with intervals. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2013. LNCS, vol. 8085, pp. 231–238. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  13. 13.
    Mucherino, A.: A pseudo de Bruijn graph representation for discretization orders for distance geometry. In: Ortuño, F., Rojas, I. (eds.) IWBBIO 2015, Part I. LNCS, vol. 9043, pp. 514–523. Springer, Heidelberg (2015) Google Scholar
  14. 14.
    Mucherino, A., Fuchs, M., Vasseur, X., Gratton, S.: Variable neighborhood search for robust optimization and applications to aerodynamics. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2011. LNCS, vol. 7116, pp. 230–237. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  15. 15.
    Mucherino, A., Lavor, C., Liberti, L.: The discretizable distance geometry problem. Optim. Lett. 6(8), 1671–1686 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.): Distance Geometry: Theory, Methods and Applications. Springer, New York (2013) Google Scholar
  17. 17.
    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

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.IRISAUniversity of Rennes 1RennesFrance

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