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The K-discretization and K-incident graphs for discretizable Distance Geometry

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Abstract

The Distance Geometry Problem (DGP) is the problem of determining whether a realization for a simple weighted undirected graph \(G=(V,E,d)\) in a given Euclidean space exists so that the distances between pairs of realized vertices u, \(v \in V\) correspond to the weights \(d_{uv}\), for each \(\{u,v\} \in E\). We focus on a special class of DGP instances, referred to as the Discretizable DGP (DDGP), and we introduce the K-discretization and the K-incident graphs for the DDGP class. The K-discretization graph is independent on the vertex order that can be assigned to V, and can be useful for discovering whether one of such orders actually exists so that the DDGP assumptions are satisfied. The use of a given vertex order allows the definition of another important graph, the K-incident graph, which is potentially useful for performing pre-processing analysis on the solution set of DDGP instances.

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Acknowledgements

We wish to thank the anonymous referees for the very fruitful comments. We are also grateful to the Brazilian research agencies FAPESP and CNPq for financial support.

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Correspondence to Antonio Mucherino.

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Abud, G., Alencar, J., Lavor, C. et al. The K-discretization and K-incident graphs for discretizable Distance Geometry. Optim Lett 14, 469–482 (2020). https://doi.org/10.1007/s11590-018-1294-2

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Keywords

  • Distance geometry
  • Vertex orders
  • Discretization
  • Combinatorial optimization