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Mathematics - Analysis | The Steiner Tree Problem - A Tour through Graphs, Algorithms, and Complexity

The Steiner Tree Problem

A Tour through Graphs, Algorithms, and Complexity

Prömel, Hans Jürgen, Steger, Angelika

Softcover reprint of the original 1st ed. 2002, VIII, 241p.

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In recent years, algorithmic graph theory has become increasingly important as a link between discrete mathematics and theoretical computer science. This textbook introduces students of mathematics and computer science to the interrelated fields of graphs theory, algorithms and complexity. No specific previous knowledge is assumed.
The central theme of the book is a geometrical problem dating back to Jakob Steiner. This problem, now called the Steiner problem, was initially of importance only within the context of land surveying. In the last decade, however, applications as diverse as VLSI-layout and the study of phylogenetic trees led to a rapid rise of interest in this problem. The resulting progress has uncovered fascinating connections between and within graph theory, the study of algorithms, and complexity theory. This single problem thus serves to bind and motivate these areas. The book's topics include: exact algorithms, computational complexity, approximation algorithms, the use of randomness, limits of approximability.
A special feature of the book is that each chapter ends with an "excursion" into some related area. These excursions reinforce the concepts and methods introduced for the Steiner problem by placing them in a broader context.

Content Level » Upper undergraduate

Keywords » Complexity - Geometric Steiner Problems - Graph - Graph theory - Graphs - Terminal Sets - algorithms

Related subjects » Algebra - Analysis

Table of contents 

1 Basics I: Graphs.- 1.1 Introduction to graph theory.- 1.2 Excursion: Random graphs.- 2 Basics II: Algorithms.- 2.1 Introduction to algorithms.- 2.2 Excursion: Fibonacci heaps and amortized time.- 3 Basics III: Complexity.- 3.1 Introduction to complexity theory.- 3.2 Excursion: More NP-complete problems.- 4 Special Terminal Sets.- 4.1 The shortest path problem.- 4.2 The minimum spanning tree problem.- 4.3 Excursion: Matroids and the greedy algorithm.- 5 Exact Algorithms.- 5.1 The enumeration algorithm.- 5.2 The Dreyfus-Wagner algorithm.- 5.3 Excursion: Dynamic programming.- 6 Approximation Algorithms.- 6.1 A simple algorithm with performance ratio 2.- 6.2 Improving the time complexity.- 6.3 Excursion: Machine scheduling.- 7 More on Approximation Algorithms.- 7.1 Minimum spanning trees in hypergraphs.- 7.2 Improving the performance ratio I.- 7.3 Excursion: The complexity of optimization problems.- 8 Randomness Helps.- 8.1 Probabilistic complexity classes.- 8.2 Improving the performance ratio II.- 8.3 An almost always optimal algorithm.- 8.4 Excursion: Primality and cryptography.- 9 Limits of Approximability.- 9.1 Reducing optimization problems.- 9.2 APX-completeness.- 9.3 Excursion: Probabilistically checkable proofs.- 10 Geometric Steiner Problems.- 10.1 A characterization of rectilinear Steiner minimum trees.- 10.2 The Steiner ratios.- 10.3 An almost linear time approximation scheme.- 10.4 Excursion: The Euclidean Steiner problem.- Symbol Index.

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