# Programming in Networks and Graphs

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

• Ulrich Derigs
Book

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

1. Front Matter
Pages I-XI
2. ### Preliminaries

1. Front Matter
Pages 1-1
2. Ulrich Derigs
Pages 2-6
3. Ulrich Derigs
Pages 7-16
4. Ulrich Derigs
Pages 17-25
3. ### The Class of General Matching Problems

1. Front Matter
Pages 27-27
2. Ulrich Derigs
Pages 28-40
3. Ulrich Derigs
Pages 41-60
4. ### Network Flow Algorithms Revisited

1. Front Matter
Pages 61-62
2. Ulrich Derigs
Pages 63-77
3. Ulrich Derigs
Pages 78-107
4. Ulrich Derigs
Pages 108-118
5. ### Bipartite Matching Problems

1. Front Matter
Pages 119-120
2. Ulrich Derigs
Pages 121-133
3. Ulrich Derigs
Pages 134-171
4. Ulrich Derigs
Pages 172-182
6. ### The 1-Matching Problem

1. Front Matter
Pages 183-184
2. Ulrich Derigs
Pages 185-199
3. Ulrich Derigs
Pages 200-253
7. ### The b-Matching Problem

1. Front Matter
Pages 255-256

### Introduction

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.

### Keywords

algorithms combinatorial optimization efficiency linear optimization optimization

#### Authors and affiliations

• Ulrich Derigs
• 1
1. 1.Universität BayreuthBayreuthGermany

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-51713-6
• Copyright Information Springer-Verlag Berlin Heidelberg 1988
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-540-18969-5
• Online ISBN 978-3-642-51713-6
• Series Print ISSN 0075-8442
• Series Online ISSN 2196-9957