Annals of Operations Research

, Volume 33, Issue 6, pp 481–499 | Cite as

A variational approach to the Steiner network problem

  • J. H. Rubinstein
  • D. A. Thomas
Section VI Steiner Tree Networks


Supposen points are given in the plane. Their coordinates form a 2n-vectorX. To study the question of finding the shortest Steiner networkS connecting these points, we allowX to vary over a configuration space. In particular, the Steiner ratio conjecture is well suited to this approach and short proofs of the casesn=4, 5 are discussed. The variational approach was used by us to solve other cases of the ratio conjecture (n=6, see [11] and for arbitraryn points lying on a circle). Recently, Du and Hwang have given a beautiful complete solution of the ratio conjecture, also using a configuration space approach but with convexity as the major idea. We have also solved Graham's problem to decide when the Steiner network is the same as the minimal spanning tree, for points on a circle and on any convex polygon, again using the variational method.


Variational Method Span Tree Variational Approach Minimal Span Tree Configuration Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© J.C. Baltzer AG, Scientific Publishing Company 1991

Authors and Affiliations

  • J. H. Rubinstein
    • 1
  • D. A. Thomas
    • 1
  1. 1.Mathematics DepartmentMelbourne UniversityParkvilleAustralia

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