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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)
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
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)
Gonçalves, D.S., Mucherino, A.: Discretization orders and efficient computation of cartesian coordinates for distance geometry. Optim. Lett. 8(7), 2111–2125 (2014)
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)
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)
Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52, 115–146 (2012)
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)
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)
Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56(1), 3–69 (2014)
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)
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)
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)
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)
Mucherino, A., Lavor, C., Liberti, L.: The discretizable distance geometry problem. Optim. Lett. 6(8), 1671–1686 (2012)
Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.): Distance Geometry: Theory, Methods and Applications. Springer, New York (2013)
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)
Acknowledgments
I am thankful to Douglas S. Gonçalves and Leo Liberti for the fruitful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Mucherino, A. (2015). Optimal Discretization Orders for Distance Geometry: A Theoretical Standpoint. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2015. Lecture Notes in Computer Science(), vol 9374. Springer, Cham. https://doi.org/10.1007/978-3-319-26520-9_25
Download citation
DOI: https://doi.org/10.1007/978-3-319-26520-9_25
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-26519-3
Online ISBN: 978-3-319-26520-9
eBook Packages: Computer ScienceComputer Science (R0)