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An integer programming approach for the search of discretization orders in distance geometry problems


Discretizable distance geometry problems (DDGPs) constitute a class of graph realization problems where the vertices can be ordered in such a way that the search space of possible positions becomes discrete, usually represented by a binary tree. Finding such vertex orders is an essential step to identify and solve DDGPs. Here we look for discretization orders that minimize an indicator of the size of the search tree. This paper sets the ground for exact solution of the optimal discretization order problem by proposing two integer programming formulations and a constraint generation procedure for an extended formulation. We discuss some theoretical aspects of the problem and numerical experiments on protein-like instances of DDGP are also reported.

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  1. 1.

    Berman, H., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T., Weissig, H., Shindyalov, I., Bourne, P.: The protein data bank. Nucleic Acids Res. 28, 235–242 (2000)

  2. 2.

    Cassioli, A., Gunluk, O., Lavor, C., Liberti, L.: Discretization vertex orders in distance geometry. Discrete Appl. Math. 197, 27–41 (2015)

  3. 3.

    Gonçalves, D.S., Mucherino, A.: Optimal partial discretization orders for discretizable distance geometry. Int. Trans. Oper. Res. 23(5), 947–967 (2016). doi:10.1111/itor.12249

  4. 4.

    Gouveia, L., Pesneau, P.: On extended formulations for the precedence constrained asymmetric traveling salesman problem. Networks 48(2), 77–89 (2006). doi:10.1002/net.20122

  5. 5.

    Grötschel, M., Jünger, M., Reinelt, G.: A Cutting plane algorithm for the linear ordering problem. Oper. Res. 32(6), 1195–1220 (1984). doi:10.1287/opre.32.6.1195. http://www.jstor.org/stable/170944. http://pubsonline.informs.org/doi/abs/10.1287/opre.32.6.1195

  6. 6.

    Grötschel, M., Jünger, M., Reinelt, G.: Facets of the linear ordering polytope. Math. Program. 33(1), 43–60 (1985). doi:10.1007/BF01582010

  7. 7.

    Lavor, C., Lee, J., John, A.L.S., Liberti, L., Mucherino, A., Sviridenko, M.: Discretization orders for distance geometry problems. Optim. Lett. 6(4), 783–796 (2012). doi:10.1007/s11590-011-0302-6

  8. 8.

    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52(1), 115–146 (2012). doi:10.1007/s10589-011-9402-6

  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), 1–17 (2008). doi:10.1111/j.1475-3995.2007.00622.x

  10. 10.

    Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56, 3–69 (2014)

  11. 11.

    Miller, C., Tucker, A., Zemlin, R.: Integer programming formulations and the travelling salesman problems. J. ACM 7, 326–329 (1960)

  12. 12.

    Mucherino, A.: MD-jeep: a software for distance geometry. http://www.antoniomucherino.it/en/mdjeep.php

  13. 13.

    Saxe, J.B.: Embeddability of weighted graphs in \(k\)-space is strongly NP-hard. In: Proceedings of 17th Allerton Conference in Communications, Control and Computing, Monticello, IL, pp. 480–489 (1979)

  14. 14.

    Tarjan, R.E.: Edge-disjoint spanning trees and depth-first search. Acta Inf. 6(2), 171–185 (1976). doi:10.1007/BF00268499

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Correspondence to Jérémy Omer.

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Omer, J., Gonçalves, D.S. An integer programming approach for the search of discretization orders in distance geometry problems. Optim Lett 14, 439–452 (2020). https://doi.org/10.1007/s11590-017-1207-9

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  • Distance geometry
  • Integer programming
  • Discretization order
  • Cutting plane