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Programming in Networks and Graphs

On the Combinatorial Background and Near-Equivalence of Network Flow and Matching Algorithms

  • Book
  • © 1988

Overview

Authors:
  1. Ulrich Derigs

    1. Universität Bayreuth, Bayreuth, Germany

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Part of the book series: Lecture Notes in Economics and Mathematical Systems (LNE, volume 300)

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Table of contents (15 chapters)

  1. Front Matter

    Pages I-XI
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  2. Preliminaries

    1. Front Matter

      Pages 1-1
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    2. Terminology

      • Ulrich Derigs
      Pages 2-6

    3. Linear Programming

      • Ulrich Derigs
      Pages 7-16

    4. Combinatorial Optimization

      • Ulrich Derigs
      Pages 17-25

  3. The Class of General Matching Problems

    1. Front Matter

      Pages 27-27
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    2. Three Cornerstone Problems

      • Ulrich Derigs
      Pages 28-40

    3. Important Subclasses

      • Ulrich Derigs
      Pages 41-60

  4. Network Flow Algorithms Revisited

    1. Front Matter

      Pages 61-62
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    2. Prologue: Two Apparently “Easier” Network Flow Problems

      • Ulrich Derigs
      Pages 63-77

    3. Approaches to Min-Cost Flow Problems

      • Ulrich Derigs
      Pages 78-107

    4. Near Equivalence of Network Flow Algorithms

      • Ulrich Derigs
      Pages 108-118

  5. Bipartite Matching Problems

    1. Front Matter

      Pages 119-120
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    2. The Cardinality Matching Problem in Bipartite Graphs

      • Ulrich Derigs
      Pages 121-133

    3. The Assignment Problem

      • Ulrich Derigs
      Pages 134-171

    4. The Hitchcock Transportation Problem

      • Ulrich Derigs
      Pages 172-182

  6. The 1-Matching Problem

    1. Front Matter

      Pages 183-184
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    2. The Cardinality Matching Problem

      • Ulrich Derigs
      Pages 185-199

    3. The Min-Cost Perfect Matching Problem

      • Ulrich Derigs
      Pages 200-253

  7. The b-Matching Problem

    1. Front Matter

      Pages 255-256
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Keywords

About this book

Network flow and matching are often treated separately in the literature and for each class a variety of different algorithms has been developed. These algorithms are usually classified as primal, dual, primal-dual etc. The question the author addresses in this work is that of the existence of a common combinatorial principle which might be inherent in all those apparently different approaches. It is shown that all common network flow and matching algorithms implicitly follow the so-called shortest augmenting path. This can be interpreted as a greedy-like decision rule where the optimal solution is built up through a sequence of local optimal solutions. The efficiency of this approach is realized by combining this myopic decision rule with an anticipant organization. The approach of this work is organized as follows. For several standard flow and matching problems the common solution procedures are first reviewed. It is then shown that they all reduce to a common basic principle, that is, they all perform the same computational steps if certain conditions are set properly and ties are broken according to a common rule. Recognizing this near-equivalence of all commonly used algorithms the question of the best method has to be modified - all methods are (only) different implementations of the same algorithm obtained by different views of the problem.

Authors and Affiliations

  • Universität Bayreuth, Bayreuth, Germany

    Ulrich Derigs

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