Network optimization problems : algorithms, applications, and complexity

405 Pages·1993·16.638 MB·English

by  Du D.-Z., Pardalos P.M. (eds.)

NETWORK
PROBLEMS
SERIES ON APPLIED MATHEMATICS
Editor-in-Chief: Frank Hwang
Associate Editors-in-Chief: Zhong-ci Shi and Kunio Tanabe
Vol. 1 International Conference on Scientific Computation
ed. T. Chan and Z.-C. Shi
P"NETWORK
PROBLEMS
ALGORITHMS, APPLICATIONS
AND COMPLEXITY
Editors
Ding-Zhu Du
Department of Computer Science
University of Minnesota
and
Institute of Applied Mathematics
Academia Sinica, Beijing
Panos M. Pardalos
Department of Industrial and Systems Engineering
University of Florida
Y J* World Scientific
wfc Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farcer Road, Singapore 9128
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Library of Congress Cataloging-in-Publication Data
Network optimization problems : algorithms, applications, and
complexity / editors, Ding-Zhu Du, Panos M. Pardalos.
p. cm. — (Series on applied mathematics; v. 2)
Includes bibliographical references.
ISBN 9810212771
1. System analysis. 2. System design. 3. Mathematical
optimization. I. Du, Dingzhu. II. Pardalos, P. M. (Panos M.),
1954- III. Series.
T57.6.N47 1993
OO3-dc20 93-16337
CIP
Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form
orby any means, electronic or mechanical, including photocopying, recording orany
information storage and retrieval system now known or to be invented, without
written permission from the Publisher.
Printed in Singapore by Utopia Press.
V
Preface
The field of networks is a lively one, both in terms of theoretical developments and
in terms of the diversity of its applications. Many problems of design and analysis of
large systems can be formulated and solved using techniques of network theory. Such
problems include communication systems, electrical networks, computer networks,
transportation, scheduling of industrial processes, facility location, and modeling of
combinatorial optimization problems.
Network theory originated many years ago, before our information age. In the
eighteenth century, Euler solved the famous Konigsberg Bridge problem and later
Kirchoff initiated the theory of electrical networks. But it was not until late last
century, when Bell invented the telephone, that many areas of network theory were
stimulated.
After the appearance of the first graph theory book (by D. Konig) in 1936, there
was tremendous development regarding the theory and applications of networks.
Hitchcock proposed the first complete algorithm for the transportation problem in
1941, Dantzig proposed the simplex algorithm for linear programming in 1947, and al
gorithms for the minimum spanning tree (Kruskal, 1956) and shortest path problems
were proposed (Prim, 1957). During the same period, the first commercial computers
became available. As it happened with many other areas of research, the fields of
computer science and networks influenced each other in many respects. In 1962 the
book by Ford and Fulkerson on "Flows in Networks" appeared. With the develop
ment of new data structure techniques and the theory of computational complexity
we entered a new era of algorithmic developments in networks.
During the second half of our century we saw major technological developments
in all areas of human endeavor and particularly in information processing. Computer
networks play a vital role in providing fast, reliable, cost-effective means of com
munication and information sharing. In addition, network techniques and computer
technology enable us to solve large-scale network models that appear in applications
such as transportation and telecommunications.
It is clear that the theory and applications of networks is so great that this book
could not give a full account and systematic treatment of the subject in its entirety. It
is our intention to introduce a number of special topics in order to show the spectrum
of recent research activities and the richness of ideas in the development of algorithms
and the applications of networks. While we were able to provide only a glimpse of this
expansive field, we felt that this glimpse would allow the reader to sense the breadth
and the depth of the field.
We would like to take the opportunity to thank the authors of the papers, the
anonymous referees, and the publisher for helping us to produce this excellent collec
tion of papers.
vi Preface
Ding-Zhu Du and Panos M. Pardalos
University of Minnesota and University of Florida
October 1992
vii
Contents
Preface v
Greedily Solvable Transportation Networks
and Edge-Guided Vertex Elimination 1
Ilan Adler and Ron Shamir
1. Introduction 1
2. Preliminaries 4
3. Vertex Elimination and Spanning Trees 7
4. Generating an Optimal Spanning Tree 11
5. Gale Certificates 12
6. Signatures 15
7. Summary 19
References 20
Networks Minimizing Length Plus
the Number of Steiner Points 23
Thomas Colthurst, Chris Cox, Joel Foisy, Hugh Howards
Kathryn Kollett, Holly Lowy, and Stephen Root
1. Introduction 23
2. Examples of Networks Minimizing Length Plus the Number of Steiner Points .24
3. Bounds on the Number of Edges Meeting at Steiner Points 32
References 35
Practical Experiences Using an Interactive Optimization
Procedure for Vehicle Scheduling 37
Joachim R. Daduna, Miodrag Mojsilovic, and Peter Schiitze
1. Introduction 37
2. Problem Formulation 38
3. Mathematical Formulation 40
4. Operational Process 43
5. Example 43
6. Results 48
7. Conclusions 49
8. Outlook 51
References 51
viii Contents
Subset Interconnection Designs: Generalizations of Spanning Trees
and Steiner Trees 53
Ding-Zhu Du and Panos M. Pardalos
1. Introduction 53
2. Multi-Phase Spanning Networks 54
3. Multi-Phase Steiner Networks 59
References 61
Polynomial and Strongly Polynomial Algorithms
for Convex Network Optimization 63
Dorit S. Hochbaum
1. Introduction 64
2. Proximity Theorems and Piecewise Linear Approximations 68
3. The Impossibility of Strongly Polynomial Algorithms 77
4. Quadratic Network Flow Problems 80
References 88
Hamiltonian Circuits for 2-Regular Interconnection Networks 93
Frank K. Hwang and Wen-Ching Winnie Li
1. Introduction 94
2. A General Approach 95
3. The Extended Double Loop Network 96
4. Semi-Torus Networks 103
5. Semi-Manhattan Networks 106
References 109
Equivalent Formulations for the Steiner Problem in Graphs Ill
Bassam N. Khoury, Panos M. Pardalos, and Donald W. Hearn
1. Introduction Ill
2. The SPG and the SPDG 112
3. Mixed Integer Formulations 113
4. Integer Formulations 114
5. Continuous Formulations 119
6. Concluding Remarks 120
References 121
Contents ix
Minimum Concave-Cost Network Flow Problems
with a Single Nonlinear Arc Cost 125
Bettina Klinz and Hoang Tuy
1. Introduction 125
2. A Parametric Method for Rank Two Quasiconcave Minimization 127
3. A Special Minimum Concave-Cost Network Flow Problem 130
4. A Strongly Polynomial Time Algorithm for SSU Networks 137
5. An Improved Algorithm for SSU Networks 138
6. Summary and Concluding Remarks 143
References 143
A Method for Solving Network Flow Problems
with General Nonlinear Arc Costs 147
Bruce W. Lamar
1. Introduction 147
2. Problem Formulation 150
3. Conversion Procedure 152
4. Numerical Examples 158
5. Summary 165
References 166
Application of Global Line Search
in Optimization of Networks 169
Jonas Mockus
1. Global Line Search 169
2. Optimization of Networks 170
3. The Optimization of High-Voltage Net of Power System 171
4. Mixed Integer Global Line Search 173
References 175
Solving Nonlinear Programs with Embedded Network Structures .... 177
Mustafa C. Pinar and Stavros A. Zenios
1. Introduction and Background 177
2. The Linear-Quadratic Penalty Algorithm for Networks with
Side Constraints and Variables 179
3. Numerical Experience 189
4. Conclusions 200
References 200
x Contents
On Algorithms for Nonlinear Dynamic Networks 203
Warren B. Powell, Elif Berkkam, and Irvin J. Lustig
1. Introduction 203
2. The NDN-X Formulation 205
3. The Transformed Problem NDN-T 207
4. Solution Algorithms for NDN-T 209
5. Numerical Results 212
6. Appendix: Calculation of the Derivatives 215
References 221
Strategic and Tactical Models and Algorithms for the Coal Industry
Under the 1990 Clean Air Act 233
Hanif D. Sherali and Quaid J. Saifee
1. Introduction 234
2. Related Models in the Literature 236
3. Formulation of a Long-Term Strategic Model 237
4. Solution Procedures 246
5. Computational Experience 249
Appendix: Modifications for a Tactical Day-to-Day 255
References 261
Multi-Objective Routing in Stochastic Evacuation Networks 263
J. MacGregor Smith
1. Problem Overview 263
2. Assumptions and Definitions 265
3. Mathematical Model 267
4. Congestion Properties 268
5. Algorithm 272
6. Example 274
7. Summary and Conclusions 280
References 280
A Simplex Method for Network Programs with
Convex Separable Piecewise Linear Costs and Its Application to
Stochastic Transshipment Problems 283
Jie Sun, K.-H. Tsai, and L. Qi
1. Introduction 284