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Brazil, M.; Rubinstein, J. H.; Thomas, D. A.; Weng, J. F.; Wormald, N.

Gradient-constrained minimum networks. III: Fixed topology. (English) Zbl 1255.90120

J. Optim. Theory Appl. 155, No. 1, 336-354 (2012).
Summary: 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).

MSC:

90C35 Programming involving graphs or networks

Keywords:

gradient constraint; Steiner trees; minimum networks; optimization

Software:

FindSteinerTree
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Full Text: DOI Link

References:

[1] Brazil, M., Rubinstein, J.H., Thomas, D.A., Weng, J.F., Wormald, N.C.: Gradient-constrained minimum networks. I. Fundamentals. J. Glob. Optim. 21, 139–155 (2001) · Zbl 1068.90605 · doi:10.1023/A:1011903210297
[2] Brazil, M., Rubinstein, J.H., Thomas, D.A., Lee, D., Weng, J.F., Wormald, N.C.: Network optimization of underground mine design. Proc. - Australas. Inst. Min. Metall. 305, 57–65 (2000)
[3] Brazil, M., Thomas, D.A., Weng, J.F.: Gradient-constrained minimum networks. II. Labelled or locally minimal steiner points. J. Glob. Optim. 42, 23–37 (2008) · Zbl 1168.90626 · doi:10.1007/s10898-007-9201-x
[4] Garey, M.R., Graham, R.L., Johnson, D.S.: The complexity of computing steiner minimal trees. SIAM J. Appl. Math. 32, 835–859 (1977) · Zbl 0399.05023 · doi:10.1137/0132072
[5] Rubinstein, J.H., Thomas, D.A., Weng, J.F.: Minimum networks for four points in space. Geom. Dedic. 93, 57–70 (2002) · Zbl 1009.05042 · doi:10.1023/A:1020389712969
[6] Smith, W.D.: How to find steiner minimal trees in euclidean d-space. Algorithmica 7, 137–177 (1992) · Zbl 0751.05028 · doi:10.1007/BF01758756
[7] Thomas, D.A., Brazil, M., Lee, D., Wormald, N.C.: Network modelling of underground mine layout: two case studies. Int. Trans. Oper. Res. 14, 143–158 (2007) · Zbl 1156.90476 · doi:10.1111/j.1475-3995.2007.00579.x
[8] Brazil, M., Thomas, D.A., Weng, J.F.: Gradient-constrained minimal Steiner trees. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 40, pp. 23–38 (1998) · Zbl 0915.05042
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