Journal of Optimization Theory and Applications

, Volume 155, Issue 1, pp 336–354 | Cite as

Gradient-Constrained Minimum Networks. III. Fixed Topology

  • M. BrazilEmail author
  • J. H. Rubinstein
  • D. A. Thomas
  • J. F. Weng
  • N. Wormald


The gradient-constrained Steiner tree problem asks for a shortest total length network interconnecting a given set of points in 3-space, where the length of each edge of the network is determined by embedding it as a curve with absolute gradient no more than a given positive value m, and the network may contain additional nodes known as Steiner points. We study the problem for a fixed topology, and show that, apart from a few easily classified exceptions, if the positions of the Steiner points are such that the tree is not minimum for the given topology, then there exists a length reducing perturbation that moves exactly 1 or 2 Steiner points. In the conclusion, we discuss the application of this work to a heuristic algorithm for solving the global problem (across all topologies).


Gradient constraint Steiner trees Minimum networks Optimization 



Research supported by the Australian Research Council, the Canadian Research Chairs Program and NSERC.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • M. Brazil
    • 1
    Email author
  • J. H. Rubinstein
    • 2
  • D. A. Thomas
    • 3
  • J. F. Weng
    • 3
  • N. Wormald
    • 4
  1. 1.Department of Electrical and Electronic EngineeringThe University of MelbourneMelbourneAustralia
  2. 2.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia
  3. 3.Department of Mechanical EngineeringThe University of MelbourneMelbourneAustralia
  4. 4.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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