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The Discretizable Distance Geometry Problem

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  • Antonio Mucherino

    (GenScale - Scalable, Optimized and Parallel Algorithms for Genomics - Inria Rennes – Bretagne Atlantique - Inria - Institut National de Recherche en Informatique et en Automatique - IRISA-D7 - GESTION DES DONNÉES ET DE LA CONNAISSANCE - IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires - UR - Université de Rennes - INSA Rennes - Institut National des Sciences Appliquées - Rennes - INSA - Institut National des Sciences Appliquées - UBS - Université de Bretagne Sud - ENS Rennes - École normale supérieure - Rennes - Inria - Institut National de Recherche en Informatique et en Automatique - Télécom Bretagne - CentraleSupélec - CNRS - Centre National de la Recherche Scientifique)

  • Carlile Lavor

    (IMECC - Instituto de Matemática, Estatística e Computação Científica [Brésil] - UNICAMP - Universidade Estadual de Campinas = University of Campinas)

  • Leo Liberti

    (LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

Abstract

We introduce the Discretizable Distance Geometry Problem in R^3 (DDGP3), which consists in a subclass of instances of the Distance Geometry Problem for which an embedding in R^3 can be found by means of a discrete search. We show that the DDGP3 is a generalization of the Discretizable Molecular Distance Geometry Problem (DMDGP), and we discuss the main differences between the two problems. We prove that the DDGP3 is NP-hard and we extend the Branch & Prune (BP) algorithm, previously used for the DMDGP, for solving instances of the DDGP3. Protein graphs may or may not be in DMDGP and/or DDGP3 depending on vertex orders and edge density. We show experimentally that as distance thresholds decrease, PDB protein graphs which fail to be in the DMDGP still belong to DDGP3, which means that they can still be solved using a discrete search.

Suggested Citation

  • Antonio Mucherino & Carlile Lavor & Leo Liberti, 2012. "The Discretizable Distance Geometry Problem," Post-Print hal-00756943, HAL.
    • Handle: RePEc:hal:journl:hal-00756943
    DOI: 10.1007/s11590-011-0358-3
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    Citations

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    Cited by:

    1. Simon J. L. Billinge & Phillip M. Duxbury & Douglas S. Gonçalves & Carlile Lavor & Antonio Mucherino, 2016. "Assigned and unassigned distance geometry: applications to biological molecules and nanostructures," 4OR, Springer, vol. 14(4), pages 337-376, December.
    2. Virginia Costa & Antonio Mucherino & Carlile Lavor & Andrea Cassioli & Luiz Carvalho & Nelson Maculan, 2014. "Discretization orders for protein side chains," Journal of Global Optimization, Springer, vol. 60(2), pages 333-349, October.
    3. Douglas S. Gonçalves & Antonio Mucherino & Carlile Lavor & Leo Liberti, 2017. "Recent advances on the interval distance geometry problem," Journal of Global Optimization, Springer, vol. 69(3), pages 525-545, November.
    4. Felipe Fidalgo & Douglas S. Gonçalves & Carlile Lavor & Leo Liberti & Antonio Mucherino, 2018. "A symmetry-based splitting strategy for discretizable distance geometry problems," Journal of Global Optimization, Springer, vol. 71(4), pages 717-733, August.
    5. Lavor, Carlile & Souza, Michael & Carvalho, Luiz M. & Gonçalves, Douglas S. & Mucherino, Antonio, 2021. "Improving the sampling process in the interval Branch-and-Prune algorithm for the discretizable molecular distance geometry problem," Applied Mathematics and Computation, Elsevier, vol. 389(C).
    6. Farid Alizadeh & Douglas Gonçalves & Nathan Krislock & Leo Liberti, 2018. "Preface: Special issue dedicated to Distance Geometry," Journal of Global Optimization, Springer, vol. 72(1), pages 1-4, September.
    7. Phil Duxbury & Carlile Lavor & Leo Liberti & Luiz Leduino Salles-Neto, 2022. "Unassigned distance geometry and molecular conformation problems," Journal of Global Optimization, Springer, vol. 83(1), pages 73-82, May.
    8. Leo Liberti, 2020. "Distance geometry and data science," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 271-339, July.
    9. Simon J. L. Billinge & Phillip M. Duxbury & Douglas S. Gonçalves & Carlile Lavor & Antonio Mucherino, 2018. "Recent results on assigned and unassigned distance geometry with applications to protein molecules and nanostructures," Annals of Operations Research, Springer, vol. 271(1), pages 161-203, December.

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