Distance Geometry in Linearizable Norms

  • Claudia D’Ambrosio
  • Leo Liberti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10589)


Distance Geometry puts the concept of distance at its center. The basic problem in distance geometry could be described as drawing an edge-weighted undirected graph in \(\mathbb {R}^K\) for some given K such that the positions for adjacent vertices have distance which is equal to the corresponding edge weight. There appears to be a lack of exact methods in this field using any other norm but \(\ell _2\). In this paper we move some first steps using the \(\ell _1\) and \(\ell _\infty \) norms: we discuss worst-case complexity, propose mixed-integer linear programming formulations, and sketch a few heuristic ideas.


Distance geometry Norms Mathematical programming 


  1. 1.
    Beeker, N., Gaubert, S., Glusa, C., Liberti, L.: Is the distance geometry problem in NP? In: Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.) Distance Geometry: Theory, Methods, and Applications. Springer, New York (2013)Google Scholar
  2. 2.
    Chiu, W.-Y., Chen, B.-S.: Mobile positioning problem in Manhattan-like urban areas: Uniqueness of solution, optimal deployment of BSs, and fuzzy implementation. IEEE Trans. Sig. Process. 57(12), 4918–4929 (2009)CrossRefMathSciNetGoogle Scholar
  3. 3.
    COIN-OR. Introduction to IPOPT: A tutorial for downloading, installing, and using IPOPT (2006)Google Scholar
  4. 4.
    Crippen, G.M.: An alternative approach to distance geometry using \(\text{ L } \infty \) distances. Discrete Appl. Math. 197, 20–26 (2015). Distance Geometry and ApplicationsCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    D’Ambrosio, C., Nannicini, G., Sartor, G.: MILP models for the selection of a small set of well-distributed points. Oper. Res. Lett. 45, 46–52 (2017)CrossRefMathSciNetGoogle Scholar
  6. 6.
    D’Ambrosio, C., Vu, K., Lavor, C., Liberti, L., Maculan, N.: New error measures and methods for realizing protein graphs from distance data. Discrete Comput. Geom. 57(2), 371–418 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dokmanić, I., Parhizkar, R., Ranieri, J., Vetterli, M.: Euclidean distance matrices: essential theory, algorithms and applications. IEEE Sig. Process. Mag. 1053–5888, 12–30 (2015)CrossRefGoogle Scholar
  8. 8.
    Erdős, P., Rényi, A.: On random graphs i. Publ. Math. (Debrecen) 6, 290–297 (1959)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Fischetti, M., Lodi, A.: Local branching. Math. Program. 98, 23–37 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Fréchet, M.: Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo 22, 1–74 (1906)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gill, P.E.: User’s guide for SNOPT version 7.2. Systems Optimization Laboratory. Stanford University, California (2006)Google Scholar
  12. 12.
    Hansen, P., Mladenović, N.: Variable neighbourhood search: principles and applications. Eur. J. Oper. Res. 130, 449–467 (2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    IBM: ILOG CPLEX 12.6 User’s Manual. IBM (2014)Google Scholar
  14. 14.
    Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56(1), 3–69 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Matousek, J.: On the distortion required for embedding finite metric spaces into normed spaces. Isr. J. Math. 93, 333–344 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Matoušek, J.: Lecture notes on metric embeddings. Technical report, ETH Zürich (2013)Google Scholar
  17. 17.
    Mehlhorn, K., Sanders, P.: Algorithms and Data Structures. Springer, Berlin (2008)zbMATHGoogle Scholar
  18. 18.
    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
  19. 19.
    Vu, K., D’Ambrosio, C., Hamadi, Y., Liberti, L.: Surrogate-based methods for black-box optimization. Int. Trans. Oper. Res. 24(3), 393–424 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Wüthrich, K.: Protein structure determination in solution by nuclear magnetic resonance spectroscopy. Science 243, 45–50 (1989)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.LIX CNRS (UMR7161)École PolytechniquePalaiseauFrance

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