The Steiner Tree Problem

1. Front Matter

   Pages I-VIII

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2. Basics I: Graphs

   • Hans Jürgen Prömel, Angelika Steger

   Pages 1-22

3. Basics II: Algorithms

   • Hans Jürgen Prömel, Angelika Steger

   Pages 23-40

4. Basics III: Complexity

   • Hans Jürgen Prömel, Angelika Steger

   Pages 41-62

5. Special Terminal Sets

   • Hans Jürgen Prömel, Angelika Steger

   Pages 63-74

6. Exact Algorithms

   • Hans Jürgen Prömel, Angelika Steger

   Pages 75-86

7. Approximation Algorithms

   • Hans Jürgen Prömel, Angelika Steger

   Pages 87-106

8. More on Approximation Algorithms

   • Hans Jürgen Prömel, Angelika Steger

   Pages 107-132

9. Randomness Helps

   • Hans Jürgen Prömel, Angelika Steger

   Pages 133-164

10. Limits of Approximability

    • Hans Jürgen Prömel, Angelika Steger

    Pages 165-190

11. Geometric Steiner Problems

    • Hans Jürgen Prömel, Angelika Steger

    Pages 191-222

12. Back Matter

    Pages 223-243

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"A very simple but instructive problem was treated by Jacob Steiner, the famous representative of geometry at the University of Berlin in the early nineteenth century. Three villages A,B ,C are to be joined by a system of roads of minimum length. " Due to this remark of Courant and Robbins (1941), a problem received its name that actually reaches two hundred years further back and should more appropriately be attributed to the French mathematician Pierre Fermat. At the end of his famous treatise "Minima and Maxima" he raised the question to find for three given points in the plane a fourth one in such a way that the sum of its distances to the given points is minimized - that is, to solve the problem mentioned above in its mathematical abstraction. It is known that Evangelista Torricelli had found a geometrical solution for this problem already before 1640. During the last centuries this problem was rediscovered and generalized by many mathematicians, including Jacob Steiner. Nowadays the term "Steiner prob­ lem" refers to a problem where a set of given points PI, . . . ,Pn have to be connected in such a way that (i) any two of the given points are joined and (ii) the total length (measured with respect to some predefined cost function) is minimized.

• Complexity
• Geometric Steiner Problems
• Graph
• Graph theory
• Graphs
• Terminal Sets
• algorithms

"The book is a very good introduction to discrete mathematics in relation to computer science, and a useful reference for those who are interested in network optimization problems." Zentralblatt MATH, Nr. 17/02

"This book is an excellent introduction to the Steiner tree problems, which starts with network Steiner trees an ends with geometric Steiner trees." Mathematical Reviews, Nr. 11/02

• Institut für Informatik, Humboldt Universität zu Berlin, Berlin, Germany

  Hans Jürgen Prömel

• Institut für Informatik, Technische Universität München, München, Germany

  Angelika Steger

Prof. Dr. Jürgen Prömel ist am Institut für Informatik der Humboldt Universität zu Berlin tätig, Prof. Dr. Angelika Steger lehrt am Institut für Informatik der TU München.

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