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Distance geometry with and without the graph

Leo Liberti, Carlile Lavor

Abstract

We survey theoretical, algorithmic, and computational results at the intersection of distance geometry problems and mathematical programming, both with and without adjacencies as part of the input. While mathematical programming methods can solve large-scale distance geometry problems with adjacencies, they are severely challenged in the absence thereof.

Distance geometry with and without the graph

Abstract

We survey theoretical, algorithmic, and computational results at the intersection of distance geometry problems and mathematical programming, both with and without adjacencies as part of the input. While mathematical programming methods can solve large-scale distance geometry problems with adjacencies, they are severely challenged in the absence thereof.

Paper Structure

This paper contains 43 sections, 2 theorems, 50 equations, 2 figures, 21 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem definitions
  3. Complexity
  4. Applications
  5. The case of proteins
  6. Basic results
  7. A mathematical programming primer
  8. MP formulations for the DGP and the UDGP
  9. The quartic formulation
  10. The unassigned quartic
  11. The system formulation
  12. The unassigned system formulation
  13. The push-and-pull formulation
  14. The unassigned push-and-pull
  15. The cycle formulation
  16. ...and 28 more sections

Key Result

Proposition 2.1

Let $\mathscr{U}=(K,n,L)$ be an instance of the UDGP, and $\alpha:[m]\to[n]\times [n]$ be an assignment. Consider the graph $G_\alpha=(V,E,d)$ where $V=[n]$, $E=\mathsf{ran}\alpha$, $d=\delta\circ\alpha^{-1}$ and the corresponding DGP instance $\mathscr{D}_\alpha=(K,G_\alpha)$. If $\mathscr{U}$ is Y

Figures (2)

  • Figure 1: The same UDGP instance gives rise to (at least) three DGP instances (two YES, one NO).
  • Figure 2: The bar plot for the graph class $\mathcal{G}$ corresponding to Table \ref{['t:Gftp']}.

Theorems & Definitions (5)

  • Proposition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 4.1