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Discretization orders for distance geometry problems

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Abstract

Given a weighted, undirected simple graph G = (V, E, d) (where \({d:E\to\mathbb{R}_+}\)), the distance geometry problem (DGP) is to determine an embedding \({x:V\to\mathbb{R}^K}\) such that \({\forall \{i,j\} \in E\;\|x_i-x_j\|=d_{ij}}\) . Although, in general, the DGP is solved using continuous methods, under certain conditions the search is reduced to a discrete set of points. We give one such condition as a particular order on V. We formalize the decision problem of determining whether such an order exists for a given graph and show that this problem is NP-complete in general and polynomial for fixed dimension K. We present results of computational experiments on a set of protein backbones whose natural atomic order does not satisfy the order requirements and compare our approach with some available continuous space searches.

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Authors and Affiliations

  1. Department of Applied Mathematics (IMECC-UNICAMP), State University of Campinas, Campinas, SP, Brazil

    Carlile Lavor

  2. IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

    Jon Lee & Maxim Sviridenko

  3. Computer Science Department, Mount Holyoke College, South Hadley, MA, USA

    Audrey Lee-St. John

  4. LIX, École Polytechnique, Palaiseau, France

    Leo Liberti

  5. CERFACS, Toulouse, France

    Antonio Mucherino

Authors
  1. Carlile Lavor
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  2. Jon Lee
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  3. Audrey Lee-St. John
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  4. Leo Liberti
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  5. Antonio Mucherino
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  6. Maxim Sviridenko
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Correspondence to Leo Liberti.

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Lavor, C., Lee, J., Lee-St. John, A. et al. Discretization orders for distance geometry problems. Optim Lett 6, 783–796 (2012). https://doi.org/10.1007/s11590-011-0302-6

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  • DOI: https://doi.org/10.1007/s11590-011-0302-6

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