Abstract
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).
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Research supported by the Australian Research Council, the Canadian Research Chairs Program and NSERC.
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Communicated by Reiner Burkard.
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Brazil, M., Rubinstein, J.H., Thomas, D.A. et al. Gradient-Constrained Minimum Networks. III. Fixed Topology. J Optim Theory Appl 155, 336–354 (2012). https://doi.org/10.1007/s10957-012-0036-3
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DOI: https://doi.org/10.1007/s10957-012-0036-3