### Abstract

The Euclidean Steiner tree problem is to find the tree with minimal Euclidean length spanning a set of fixed points in the plane, allowing the addition of auxiliary points to the set (Steiner points). The problem is NP-hard, so polynomial-time heuristics are desired. We present two such heuristics, both of which utilize an efficient method for computing a locally optimal tree with a given topology. The first systematically inserts Steiner points between edges of the minimal spanning tree meeting at angles less than 120 degrees, performing a local optimization at the end. The second begins by finding the Steiner tree for three of the fixed points. Then, at each iteration, it introduces a new fixed point to the tree, connecting it to each possible edge by inserting a Steiner point, and minimizes over all connections, performing a local optimization for each. We present a variety of test cases that demonstrate the strengths and weaknesses of both algorithms.

Original language | English (US) |
---|---|

Pages (from-to) | 95-106 |

Number of pages | 12 |

Journal | Journal of Global Optimization |

Volume | 13 |

Issue number | 1 |

State | Published - 1998 |

### Fingerprint

### Keywords

- Euclidean Steiner tree
- Interior-point algorithm

### ASJC Scopus subject areas

- Management Science and Operations Research
- Global and Planetary Change
- Applied Mathematics
- Control and Optimization

### Cite this

*Journal of Global Optimization*,

*13*(1), 95-106.

**Two Heuristics for the Euclidean Steiner Tree Problem.** / Dreyer, Derek R.; Overton, Michael L.

Research output: Contribution to journal › Article

*Journal of Global Optimization*, vol. 13, no. 1, pp. 95-106.

}

TY - JOUR

T1 - Two Heuristics for the Euclidean Steiner Tree Problem

AU - Dreyer, Derek R.

AU - Overton, Michael L.

PY - 1998

Y1 - 1998

N2 - The Euclidean Steiner tree problem is to find the tree with minimal Euclidean length spanning a set of fixed points in the plane, allowing the addition of auxiliary points to the set (Steiner points). The problem is NP-hard, so polynomial-time heuristics are desired. We present two such heuristics, both of which utilize an efficient method for computing a locally optimal tree with a given topology. The first systematically inserts Steiner points between edges of the minimal spanning tree meeting at angles less than 120 degrees, performing a local optimization at the end. The second begins by finding the Steiner tree for three of the fixed points. Then, at each iteration, it introduces a new fixed point to the tree, connecting it to each possible edge by inserting a Steiner point, and minimizes over all connections, performing a local optimization for each. We present a variety of test cases that demonstrate the strengths and weaknesses of both algorithms.

AB - The Euclidean Steiner tree problem is to find the tree with minimal Euclidean length spanning a set of fixed points in the plane, allowing the addition of auxiliary points to the set (Steiner points). The problem is NP-hard, so polynomial-time heuristics are desired. We present two such heuristics, both of which utilize an efficient method for computing a locally optimal tree with a given topology. The first systematically inserts Steiner points between edges of the minimal spanning tree meeting at angles less than 120 degrees, performing a local optimization at the end. The second begins by finding the Steiner tree for three of the fixed points. Then, at each iteration, it introduces a new fixed point to the tree, connecting it to each possible edge by inserting a Steiner point, and minimizes over all connections, performing a local optimization for each. We present a variety of test cases that demonstrate the strengths and weaknesses of both algorithms.

KW - Euclidean Steiner tree

KW - Interior-point algorithm

UR - http://www.scopus.com/inward/record.url?scp=0002218967&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002218967&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0002218967

VL - 13

SP - 95

EP - 106

JO - Journal of Global Optimization

JF - Journal of Global Optimization

SN - 0925-5001

IS - 1

ER -