Two Heuristics for the Euclidean Steiner Tree Problem

Derek R. Dreyer, Michael L. Overton

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)95-106
Number of pages12
JournalJournal of Global Optimization
Volume13
Issue number1
StatePublished - 1998

Fingerprint

Steiner Point
Steiner Tree Problem
heuristics
Euclidean
Local Optimization
Fixed point
Heuristics
Minimal Spanning Tree
Computational complexity
Steiner Tree
Topology
Polynomials
topology
Polynomial time
NP-complete problem
Minimise
Iteration
Angle
Computing
Demonstrate

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

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

In: Journal of Global Optimization, Vol. 13, No. 1, 1998, p. 95-106.

Research output: Contribution to journalArticle

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