Flows in Networks

Lester Randolph Ford; D. R. Fulkerson

Book

Flows in Networks

Lester Randolph Ford and D. R. Fulkerson
Preface by: Robert G. Bland , James B. Orlin , Robert G. Bland and James B. Orlin

Language: English

Published/Copyright: 2024

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About this book

A landmark work that belongs on the bookshelf of every researcher working with networks

In this classic book, first published in 1962, L. R. Ford, Jr., and D. R. Fulkerson set the foundation for the study of network flow problems. The models and algorithms introduced in Flows in Networks are used widely today in the fields of transportation systems, manufacturing, inventory planning, image processing, and Internet traffic. The techniques presented by Ford and Fulkerson spurred the development of powerful computational tools for solving and analyzing network flow models, and also furthered the understanding of linear programming. In addition, the book helped illuminate and unify results in combinatorial mathematics while emphasizing proofs based on computationally efficient construction. With an incisive foreword by Robert Bland and James Orlin, Flows in Networks is rich with insights that remain relevant to current research in engineering, management, and other sciences.

Author / Editor information

L. R. Ford, Jr., (1927–2017) was a researcher at CEIR Inc. and the Rand Corporation. D. R. Fulkerson (1924–1976) was a mathematician at the Rand Corporation.

Reviews

“[Flows in Networks] should . . . be of great value to the expert and a standard reference source for many years to come.”—H. J. Ryser, Management Science

“The book stands as the principal work on network flow theory. Its authors have performed almost as great a service in preparing this volume for publication as they did in originally developing much of its contents.”—Ronald A. Howard, Proceedings of the IEEE

“The book should be of value not only to those interested in linear programming but also those who are concerned with graph theory.”—Arthur Ziffer, Physics Today

“The book is a natural meeting ground for persons interested in communication engineering or combinatorial mathematics.”—Journal of Data Management

“The book is a very welcome addition to the literature and should be of extreme value to anyone interested in operations research, communication theory, or combinatorial mathematics.”—A. Newhouse, Computing Review

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Frontmatter
i

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CONTENTS
vii

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Foreword to the 2010 edition
ix

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PREFACE
xvii

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ACKNOWLEDGMENTS
xix

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CHAPTER I STATIC MAXIMAL FLOW
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Introduction
1

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1. Networks
2

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2. Flows in networks
4

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3. Notation
9

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4. Cuts
10

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5. Maximal flow
11

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6. Disconnecting sets and cuts
14

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7. Multiple sources and sinks
15

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8. The labeling method for solving maximal flow problems
17

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9. Lower bounds on arc flows
22

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10. Flows in undirected and mixed networks
23

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11. Node capacities and other extensions
23

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12. Linear programming and duality principles
26

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13. Maximal flow value as a function of two arc capacities
30

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References
35

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CHAPTER II FEASIBILITY THEOREMS AND COMBINATORIAL APPLICATIONS
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Introduction
36

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1. A supply-demand theorem
36

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2. A symmetric supply-demand theorem
42

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3. Circulation theorem
50

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4. The König-Egerváry and Menger graph theorems
53

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5. Construction of a maximal independent set of admissible cells
55

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6. A bottleneck assignment problem
57

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7. Unicursal graphs
59

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8. Dilworth's chain decomposition theorem for partially ordered sets
61

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9. Minimal number of individuals to meet a fixed schedule of tasks
64

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10. Set representatives
67

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11. The subgraph problem for directed graphs
75

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12. Matrices composed of O's and l's
79

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References
91

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CHAPTER III MINIMAL COST FLOW PROBLEMS
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Introduction
93

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1. The Hitchcock problem
95

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2. The optimal assignment problem [56, 57, 60, 61, 68, 69]
111

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3. The general minimal cost flow problem
113

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4. Equivalence of Hitchcock and minimal cost flow problems
127

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5. A shortest chain algorithm
130

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6. The minimal cost supply-demand problem: non-negative directed cycle costs
134

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7. The warehousing problem
137

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8. The caterer problem
140

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9. Maximal dynamic flow
142

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10. Project cost curves
151

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11. Constructing minimal cost circulations [28]
162

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References
169

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CHAPTER IV MULTI-TERMINAL MAXIMAL FLOWS
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Introduction
173

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1. Forests, trees, and spanning subtrees
173

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2. Realization conditions
176

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3. Equivalent networks
177

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4. Network synthesis
187

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References
191

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INDEX
193

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Publishing information

Pages and Images/Illustrations in book

eBook published on:

December 3, 2024

eBook ISBN:

9780691273457

Pages and Images/Illustrations in book

https://doi.org/10.1515/9780691273457

eBook ISBN: 9780691273457

Audience(s) for this book

For a non-specialist adult audience

Flows in Networks

Overview

About this book

Author / Editor information

Reviews

Topics

Table of contents

Frontmatter

CONTENTS

Foreword to the 2010 edition

PREFACE

ACKNOWLEDGMENTS

Introduction

1. Networks

2. Flows in networks

3. Notation

4. Cuts

5. Maximal flow

6. Disconnecting sets and cuts

7. Multiple sources and sinks

8. The labeling method for solving maximal flow problems

9. Lower bounds on arc flows

10. Flows in undirected and mixed networks

11. Node capacities and other extensions

12. Linear programming and duality principles

13. Maximal flow value as a function of two arc capacities

References

Introduction

1. A supply-demand theorem

2. A symmetric supply-demand theorem

3. Circulation theorem

4. The König-Egerváry and Menger graph theorems

5. Construction of a maximal independent set of admissible cells

6. A bottleneck assignment problem

7. Unicursal graphs

8. Dilworth's chain decomposition theorem for partially ordered sets

9. Minimal number of individuals to meet a fixed schedule of tasks

10. Set representatives

11. The subgraph problem for directed graphs

12. Matrices composed of O's and l's

References

Introduction

1. The Hitchcock problem

2. The optimal assignment problem [56, 57, 60, 61, 68, 69]

3. The general minimal cost flow problem

4. Equivalence of Hitchcock and minimal cost flow problems

5. A shortest chain algorithm

6. The minimal cost supply-demand problem: non-negative directed cycle costs

7. The warehousing problem

8. The caterer problem

9. Maximal dynamic flow

10. Project cost curves

11. Constructing minimal cost circulations [28]

References

Introduction

1. Forests, trees, and spanning subtrees

2. Realization conditions

3. Equivalent networks

4. Network synthesis

References

INDEX

Bibliographic data