An overview of exact algorithms for the Euclidean Steiner tree problem in <i>n</i>‐space

Fampa, Marcia; Lee, Jon; Maculan, Nelson

doi:10.1111/itor.12207

Cited by 15 publications

(16 citation statements)

References 22 publications

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“…To address the identified shortcomings of the methodology of [16] for smoothing square roots, motivated by developing tractable mixed-integer non-linearoptimization models for the Euclidean Steiner Problem (see [14]), [10,11] developed a virtuous method. On the interval [0, δ], they fit a homogeneous cubic, to match the function value of the square root at the endpoints, and matching the first and second derivatives at w = δ.…”

Section: Introductionmentioning

confidence: 99%

Virtuous smoothing for global optimization

Lee

Skipper

2017

J Glob Optim

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In the context of global optimization and mixed-integer non-linear programming, generalizing a technique of D'Ambrosio, Fampa, Lee and Vigerske for handling the square-root function, we develop a virtuous smoothing method, using cubics, aimed at functions having some limited non-smoothness. Our results pertain to root functions (w p with 0 < p < 1) and their increasing concave relatives. We provide (i) a sufficient condition (which applies to functions more general than root functions) for our smoothing to be increasing and concave, (ii) a proof that when p = 1/q for integers q ≥ 2, our smoothing lower bounds the root function, (iii) substantial progress (i.e., a proof for integers 2 ≤ q ≤ 10, 000) on the conjecture that our smoothing is a sharper bound on the root function than the natural and simpler "shifted root function", and (iv) for all root functions, a quantification of the superiority (in an average sense) of our smoothing versus the shifted root function near 0.

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Section: Introductionmentioning

confidence: 99%

Virtuous smoothing for global optimization

Lee

Skipper

2017

J Glob Optim

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“…Note that DB 7 = DB. 0.6000 DB 4 6.0525 DB 5 26.2374 DB 6 51.3653 DB 7 = DB 62.8098 Figure 7: DB k for k = 3, 4, ..., 7, d = 6 and n = 100.…”

Section: Computational Resultsmentioning

confidence: 99%

“…However, no analytical method can exist for d ≥ 3 [1]. Furthermore, no numerical approximation seems to be able to solve instances with more than 15-20 terminals [6]. It is therefore essential to develop good quality heuristics for d ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

Steiner Tree Heuristic in the Euclidean d-Space Using Bottleneck Distances

Lorenzen

Winter

2016

Experimental Algorithms

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Some of the most efficient heuristics for the Euclidean Steiner minimal trees in the d-dimensional space, d ≥ 2, use Delaunay tessellations and minimum spanning trees to determine small subsets of geometrically close terminals. Their low-cost Steiner trees are determined and concatenated in a greedy fashion to obtain low cost trees spanning all terminals. The weakness of this approach is that obtained solutions are topologically related to minimum spanning trees. To obtain better solutions, bottleneck distances are utilized to determine good subsets of terminals without being constrained by the topologies of minimum spanning trees. Computational experiments show a significant solution quality improvement.

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“…• R Si is linked to S i by an edge of index 2 * i + 2. Figure 4 shows the process of building topology (3,5).…”

Section: Topology Reorganizationmentioning

confidence: 99%

“…In [4], the authors proposed improvements to Smith's algorithm and a new branch enumeration algorithm in a sense close to the approach used by GeoSteiner, the powerful solver for planar problems. An overview of exact algorithms in d-space is given in [3]. The three common approaches to improve Smith's algorithm are the choice of a better fathoming criterion, to decide the order in which regular nodes are inserted, and the computation of the optimization process.…”

Section: Introductionmentioning

confidence: 99%

Reorganizing topologies of Steiner trees to accelerate their eliminations

Grodet

Tsuchiya

2019

Discrete Math. Algorithm. Appl.

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We describe a technique to reorganize topologies of Steiner trees by exchanging neighbors of adjacent Steiner points. We explain how to use the systematic way of building trees, and therefore topologies, to find the correct topology after nodes have been exchanged. Topology reorganizations can be inserted into the enumeration scheme commonly used by exact algorithms for the Euclidean Steiner tree problem in d-space, providing a method of improvement different than the usual approaches. As an example, we show how topology reorganizations can be used to dynamically change the exploration of the usual branch-and-bound tree when two Steiner points collide during the optimization process. We also turn our attention to the erroneous use of a pre-optimization lower bound in the original algorithm and give an example to confirm its usage is incorrect. In order to provide numerical results on correct solutions, we use planar equilateral points to quickly compute this lower bound, even in dimensions higher than two. Finally, we describe planar twin trees, identical trees yielded by different topologies, whose generalization to higher dimensions could open a new way of building Steiner trees.

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An overview of exact algorithms for the Euclidean Steiner tree problem in n‐space

Cited by 15 publications

References 22 publications

Virtuous smoothing for global optimization

Virtuous smoothing for global optimization

Steiner Tree Heuristic in the Euclidean d-Space Using Bottleneck Distances

Reorganizing topologies of Steiner trees to accelerate their eliminations

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