Numerical Calibration of Steiner trees

Abstract

In this paper we propose a variational approach to the Steiner tree problem, which is based on calibrations in a suitable algebraic environment for polyhedral chains which represent our candidates. This approach turns out to be very efficient from numerical point of view and allows to establish whether a given Steiner tree is optimal. Several examples are provided.

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Notes

  1. 1.

    The endpoints of each segment \(\sigma _i\) will be denoted by \(x_i\) and \(y_i\in \mathbb {R}^d\) (with orientation from \(x_i\) to \(y_i\)).

  2. 2.

    When \(d>3\), the curl above stands for the exterior derivative of the 1-form associated with \(M_{(j)}\).

  3. 3.

    The existence of such a decomposition is known for one-dimensional currents (see [2]) and consequently for one-dimensional polyhedral G-chains (see [6]). Heuristically, for a finite number of segments (as in the case of polyhedral G-chains), one may explicitly perform the decomposition following the path of a fixed coefficient. This path has to close up to prevent the formation of boundary.

  4. 4.

    Notice that the group G is independent from the position of the points \(x_1,\ldots ,x_n\in \mathbb {R}^d\).

  5. 5.

    We use the notation \(a\vee b:=\max \{a,b\}\) and \(a\wedge b:=\min \{a,b\}\).

References

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Acknowledgements

We would like to thank the MmgTools team (see https://github.com/MmgTools/mmg): Charles Dapogny, Cécile Dobrzynski, Pascal Frey and Algiane Froehly) for providing the remeshing software required to generate constraint meshes. Édouard Oudet gratefully acknowledges the support of the ANR, through the projects COMEDIC, GEOMETRYA, PGMO and OPTIFORM.

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Correspondence to Bozhidar Velichkov.

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Massaccesi, A., Oudet, E. & Velichkov, B. Numerical Calibration of Steiner trees. Appl Math Optim 79, 69–86 (2019). https://doi.org/10.1007/s00245-017-9421-5

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Keywords

  • Calibration
  • Minimal surface
  • Steiner tree problem

Mathematics Subject Classification

  • 49J45
  • 35R35
  • 49M05
  • 35J25