### 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) |
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Pages (from-to) | 95-106 |

Number of pages | 12 |

Journal | Journal of Global Optimization |

Volume | 13 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1998 |

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### Keywords

- Euclidean Steiner tree
- Interior-point algorithm

### ASJC Scopus subject areas

- Computer Science Applications
- Management Science and Operations Research
- Control and Optimization
- Applied Mathematics

### Cite this

*Journal of Global Optimization*,

*13*(1), 95-106. https://doi.org/10.1023/A:1008285504599