## Description

This volume is devoted to the assimilation of the rich field of intriguing analyses and the consolidation of the fragments. A section has been given to each of the three major areas of interest which have emerged. The first concerns the Euclidean Steiner Problem, historically the original Steiner tree problem proposed by Jarník and Kössler in 1934. The second deals with the Steiner Problem in Networks, which was propounded independently by Hakimi and Levin and has enjoyed the most prolific research amongst the three areas. The Rectilinear Steiner Problem, introduced by Hanan in 1965, is discussed in the third part. Additionally, a forth section has been included, with chapters discussing areas where the body of results is still emerging.

The collaboration of three authors with different styles and outlooks affords individual insights within a cohesive whole.

## Table of Contents

**Euclidean Steiner Problem.**Introduction.

*Historical Background. Some Basic Notions. Some Basic Properties. Full Steiner Trees. Steiner Hulls and Decompositions. The Number of Steiner Topologies. Computational Complexity. Physical Models. References.*Exact Algorithms.

*The Melzak Algorithm. A Linear-Time FST Algorithm. Two Ideas on the Melzak Algorithm. A Numerical Algorithm. Pruning. The GEOSTEINER Algorithm. The Negative Edge Algorithm. The Luminary Algorithm. References.*The Steiner Ratio.

*Lower Bounds of &rgr;. The Small n Case. The Variational Approach. The Steiner Ratio Conjecture as a Maximin Problem. Critical Structures. A Proof of the Steiner Ratio Conjecture. References.*Heuristics.

*Minimal Spanning Trees. Improving the MST. Greedy Trees. An Annealing Algorithm. A Partitioning Algorithm. Few's Algorithms. A Graph Approximation Algorithm. k-Size Quasi-Steiner Trees. Other Heuristics. References.*Special Terminal-Sets.

*Four Terminals. Cocircular Terminals. Co-path Terminals. Terminals on Lattice Points. Two Related Results. References.*Generalizations.

*d-Dimensional Euclidean Spaces. Cost of Edge. Terminal Clusters and New Terminals. k-SMT. Obstacles. References.*

**Steiner Problem in Networks.**Introduction.

*Applications. Definitions. Trivial Special Cases. Problem Reformulations. Complexity. References.*Reductions.

*Exclusion Tests. Inclusion Tests. Integration of Tests. Effectiveness of Reductions. References.*Exact Algorithms.

*Spanning Tree Enumeration Algorithm. Degree-Constrained Tree Enumeration Algorithm. Topology Enumeration Algorithm. Dynamic Programming Algorithm. Branch-and-Bound Algorithm. Mathematical Programming Formulations. Linear Relaxations. Lagrangean Relaxations. Benders' Decomposition Algorithm. Set Covering Algorithm. Summary and Computational Experience. References.*Heuristics.

*Path Heurisitics. Tree Heuristics. Vertex Heuri*

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AT&T Bell Laboratories, Murray Hill, NJ, USA
University of Virginia, Thornton Hall, VA, USA
University of Copenhagen, Copenhagen, Denmark

## Details

- No. of pages:
- 336

- Language:
- English

- Copyright:
- © 1992

- Published:
- 20th October 1992

- Imprint:
- North Holland

- eBook ISBN:
- 9780080867939

- Print ISBN:
- 9780444890986

## About the authors

### F.K. Hwang

#### Affiliations and Expertise

### D.S. Richards

#### Affiliations and Expertise

### P. Winter

#### Affiliations and Expertise