Numerical Calibration of Steiner trees


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  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 \(a\wedge b:=\min \{a,b\}\).


  1. 1.

    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vision 40(1), 120–145 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Federer, H.: Geometric Measure Theory. Springer, New York (1969)

    Google Scholar 

  3. 3.

    Gilbert, E.N., Pollak, H.O.: Steiner minimal trees. SIAM J. Appl. Math. 16, 1–29 (1968)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Ivanov, A.O., Tuzhilin, A.A.: Minimal networks. The Steiner problem and its generalizations. CRC Press, Boca Raton (1994)

    Google Scholar 

  5. 5.

    Karp, R.M.: Reducibility among combinatorial problems, Complexity of computer computations In: Proceedings Symposium, IBM Thomas J. Watson Res. Center, pp. 85–103 Yorktown Heights, NY (1972)

  6. 6.

    Marchese, A., Massaccesi, A.: The Steiner tree problem revisited through rectifiable \(G\)-currents. Adv. Calculus Variations 9(1), 19–39 (2016)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Warme, D.M., Winter, P., Zachariasen, M.: GeoSteiner 3.1 Department of Computer Science. University of Copenhagen (DIKU), Copenhagen (2001)

    Google Scholar 

Download references


We would like to thank the MmgTools team (see 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.

Author information



Corresponding author

Correspondence to Bozhidar Velichkov.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Massaccesi, A., Oudet, E. & Velichkov, B. Numerical Calibration of Steiner trees. Appl Math Optim 79, 69–86 (2019).

Download citation


  • Calibration
  • Minimal surface
  • Steiner tree problem

Mathematics Subject Classification

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