An overview of exact algorithms for the Euclidean Steiner tree problem inn-space

This paper presents a genetic algorithm for the Euclidean Steiner tree problem. This is an optimization problem whose objective is to obtain a minimum length tree to interconnect a set of fixed points, and for this purpose to be achieved, new auxiliary points, called Steiner points, can be added. The proposed heuristic uses a genetic algorithm to manipulate spanning trees, which are then transformed into Steiner trees by inserting and repositioning the Steiner points. Greedy genetic operators and evolutionary strategies are tested. Results of numerical experiments for benchmark library problem (OR-Library) are presented and discussed.This paper presents a genetic algorithm for the Euclidean Steiner tree problem. This is an optimization problem whose objective is to obtain a minimum length tree to interconnect a set of fixed points, and for this purpose to be achieved, new auxiliary points, called Steiner points, can be added. The proposed heuristic uses a genetic algorithm to manipulate spanning trees, which are then transformed into Steiner trees by inserting and repositioning the Steiner points. Greedy genetic operators and evolutionary strategies are tested. Results of numerical experiments for benchmark library problem (OR-Library) are presented and discussed.

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On a nonconvex MINLP formulation of the Euclidean Steiner tree problem in n-space: missing proofs

Optimization Letters ◽

10.1007/s11590-018-1295-1 ◽

2018 ◽

Vol 14 (2) ◽

pp. 409-415 ◽

Cited By ~ 1

Author(s):

Claudia D’Ambrosio ◽

Marcia Fampa ◽

Jon Lee ◽

Stefan Vigerske

Keyword(s):

Steiner Tree ◽

Steiner Tree Problem ◽

Euclidean Steiner Tree Problem ◽

Missing Proofs

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Reorganizing topologies of Steiner trees to accelerate their eliminations

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500032 ◽

2019 ◽

Vol 12 (01) ◽

pp. 2050003

Author(s):

Aymeric Grodet ◽

Takuya Tsuchiya

Keyword(s):

Lower Bound ◽

Higher Dimensions ◽

Exact Algorithms ◽

Steiner Trees ◽

Steiner Tree Problem ◽

Original Algorithm ◽

Steiner Points ◽

Euclidean Steiner Tree Problem ◽

Correct Topology ◽

Enumeration Scheme

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 [Formula: see text]-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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