Theory of Linear and Integer Programming

485 Pages·1986·6.35 MB·English

by  Alexander Schrijver

THEORY OF LINEAR AND
INTEGER PROGRAMMING
WILEY-INTERSCIENCE
SERIES IN DISCRETE MATHEMATICS
ADVISORY EDITORS
Ronald L. Graham
AT & T Bell Laboratories, Murray Hill, New Jersey, U.S.A.
Jan Karel Lenstra
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Graham, Rothschild, and Spencer
RAMSEY THEORY
Tucker
APPLIED COMBINATORICS
Pless
INTRODUCTION TO THE THEORY OF ERROR-CORRECTING CODES
Nemirovsky and Yudin
PROBLEM COMPLEXITY AND METHOD EFFICIENCY IN OPTIMIZATION
(Translated by E. R. Dawson)
Goulden and Jackson
COMBINATORIAL ENUMERATION
Gondran and Minoux
GRAPHS AND ALGORITHMS
(Translated by S. Vajda)
Fishburn
INTERVAL ORDERS AND INTERVAL GRAPHS
Tomescu
PROBLEMS IN COMBINATORICS AND GRAPH THEORY
(Translated by R. A. Melter)
Palmer
GRAPHICAL EVOLUTION
Lawler, Lenstra, Rinnooy Kan, and Shmoys
THE TRAVELING SALESMAN PROBLEM
Schrijver
THEORY OF LINEAR AND INTEGER PROGRAMMING
THEORY OF LINEAR AND
INTEGER PROGRAMMING
ALEXANDER SCHRIJVER
Department of Econometrics,
Tilburg University
and
Centrum voor Wiskunde en Informatica,
Amsterdam
A Wiley-Interscience Publication
JOHN WILEY & SONS
Chichester New York Brisbane Toonto- Singapore
A. Schrijver,
Department of Econometrics,
Tilburg University,
P.O. Box 90153,
5000 LE Tilburg,
The Netherlands.
Copyright Q 1986 by John Wiley & Sons Ltd.
Reprinted with corrections October 1987
All rights reserved.
No part of this book may be reproduced by any means, or
transmitted, or translated into a machine language without
the written permission of the publisher.
Library of Congress Cataloging-in-Publication Data:
Schrijver, Alexander.
Theory of linear and integer programming.
(Wiley-Interscience series in discrete mathematics)
'A Wiley-Interscience publication.'
Bibliography: p.
Includes index.
1. Linear programming. 2. Integer programming.
1. Title. H. Series.
T57.74.S53 1986 519.7'2 85-12314
ISBN 0 471 90854 1
British Library Cataloguing in Publication Data:
Schrijver, Alexander
Theory of linear and integer programming.
1. Linear programming 2. Integer programming
1. Title
519.7'2 T57.74
ISBN 0 471 90854 1
Printed and Bound in Great Britain by Anchor Brendan Ltd, Tiptree,. Essex
Preface
There exist several excellent books on linear and integer programming. Yet, I
did not feel it superfluous to write the present book. Most of the existing books
focus on the, very important, algorithmic side of linear and integer programming.
The emphasis of this book is on the more theoretical aspects, and it aims at
complementing the more practically oriented books.
Another reason for writing this book is that during the last few years several
interesting new results have been obtained, which are not yet all covered by
books: Lovasz's basis reduction method, Khachiyan's ellipsoid method and
Karmarkar's method for linear programming, Borgwardt's analysis of the
average speed of the simplex method, Tardos' and Megiddo's algorithms for
linear programming, Lenstra's algorithm for integer linear programming,
Seymour's decomposition theorem for totally unimodular matrices, and the
theory of total dual integrality.
Although the emphasis is on theory, this book does not exclude algorithms.
This is motivated not only by the practical importance of algorithms, but also
by the fact that the complexity analysis of problems and algorithms has become
more and more a theoretical topic as well. In particular, the study of polynomial-
time solvability has led to interesting theory. Often the polynomial-time
solvability of a certain problem can be shown theoretically (e.g. with the ellipsoid
method); such a proof next serves as a motivation for the design of a method
which is efficient in practice. Therefore we have included a survey of methods
known for linear and integer programming, together with a brief analysis of
their running time. Our descriptions are meant for a quick understanding of
the method, and might be, in many cases, less appropriate for a direct
implementation.
The book also arose as a prerequisite for the forthcoming book Polyhedral
Combinatorics, dealing with polyhedral (i.e. linear programming) methods in
combinatorial optimization. Dantzig, Edmonds, Ford, Fulkerson, and Hoffman
have pioneered the application of polyhedral methods to combinatorial optimiz-
ation, and now combinatorial optimization is dissolubly connected to (integer)
linear programming. The book Polyhedral Combinatorics describes these connec-
tions, which heavily lean on results and methods discussed in the present book.
For a better understanding, and to make this book self-contained, we have
illustrated some of the results by combinatorial applications.
vi Preface
Several friends and colleagues have helped and inspired me in preparing this
book. It was Cor Baayen who stimulated me to study discrete mathematics,
especially combinatorial optimization, and who advanced the idea of compiling
a monograph on polyhedral methods in combinatorial optimization. During
leaves of absence spent in Oxford and Szeged (Hungary) I enjoyed the hospitality
of Paul Seymour and Laci Lovasz. Their explanations and insights have helped
me considerably in understanding polyhedral combinatorics and integer linear
programming. Concurrently with the present book, I was involved with Martin
Grotschel and Laci Lovasz in writing the book The Ellipsoid Method and
Combinatorial Optimization (Springer-Verlag, Heidelberg). Although the plans
of the two books are distinct, there is some overlap, which has led to a certain
cross-fertilization. I owe much to the pleasant cooperation with my two co-
authors. Also Bob Bixby, Bill Cook, Bill Cunningham, Jack Edmonds, Werner
Fenchel, Bert Gerards, Alan Hoffman, Antoon Kolen, Jaap Ponstein, Andras
Sebo, Eva Tardos, Klaus Trumper and Laurence Wolsey have helped me by
pointing out to me information and ideas relevant to the book, or by reading
and criticizing parts of the manuscript. The assistance of the staff of the library
of the Mathematical Centre, in particular of Carin Klompen, was important in
collecting many ancient articles indispensable for composing the historical
surveys.
Thanks are due to all of them. I also acknowledge hospitality and/or financial
support given by the following institutions and organizations: the Mathematical
Centre/Centrum voor Wiskunde en Informatica, the Netherlands organization
for the advancement of pure research Z.W.O., the University of Technology
Eindhoven, the Bolyai Institute of the Attila Jozsef University in Szeged, the
University of Amsterdam, Tilburg University, and the Institut fur Okonometrie
and Operations Research of the University of Bonn.
Finally, I am indebted to all involved in the production of this book. It has
been a pleasure to work with Ian McIntosh and his colleagues of John Wiley &
Sons Limited. In checking the galley proofs, Theo Beekman, Jeroen van den Berg,
Bert Gerards, Stan van Hoesel, Cor Hurkens, Hans Kremers, Fred Nieuwland,
Henk Oosterhout, Joke Sterringa, Marno Verbeek, Hein van den Wildenberg,
and Chris Wildhagen were of great assistance, and they certainly cannot be
blamed for any surviving errors.
ALEXANDER SCHRIJVER
Contents
Introduction and preliminaries
1 1
1.1 Introduction, 1
1.2 General preliminaries, 3
1.3 Preliminaries from linear algebra, matrix theory, and Euclidean geometry, 4
1.4 Some graph theory, 8
2 Problems, algorithms, and complexity 14
2.1 Letters, words, and sizes, 15
2.2 Problems, 15
2.3 Algorithms and running time, 16
2.4 Polynomial algorithms, 17
2.5 The classes 9, .K9, and co-.N'9, 18
2.6 .K9-complete problems, 20
Some historical notes, 21
PART I: LINEAR ALGEBRA
25
3 Linear algebra and complexity 27
3.1 Some theory, 27
3.2 Sizes and good characterizations, 29
3.3 The Gaussian elimination method, 31
3.4 Iterative methods, 36
Notes on linear algebra
38
Historical notes, 38
Further notes on linear algebra, 40
PART II: LATTICES AND LINEAR DIOPHANTINE
EQUATIONS 43
4 Theory of lattices and linear diophantine equations 45
4.1 The Hermite normal form, 45
4.2 Uniqueness of the Hermite normal form, 48
4.3 Unimodular matrices, 48
4.4 Further remarks, 50
vii
viii Contents
5 Algorithms for linear diophantine equations 52
5.1 The Euclidean algorithm, 52
5.2 Sizes and good characterizations, 54
5.3 Polynomial algorithms for Hermite normal forms and systems of linear
diophantine equations, 56
6 Diophantine approximation and basis reduction 60
6.1 The continued fraction method, 60
6.2 Basis reduction in lattices, 67
6.3 Applications of the basis reduction method, 71
Notes on lattices and linear diophantine equations 76
Historical notes, 76
Further notes on lattices and linear diophantine equations, 82
PART III: POLYHEDRA, LINEAR INEQUALITIES,
AND LINEAR PROGRAMMING
83
7 Fundamental concepts and results on polyhedra, linear inequalities,
and linear programming
85
7.1 The Fundamental theorem of linear inequalities, 85
7.2 Cones, polyhedra, and polytopes, 87
7.3 Farkas' lemma and variants, 89
7.4 Linear programming, 90
7.5 LP-duality geometrically, 92
7.6 Affine form of Farkas' lemma, 93
7.7 Caratheodory's theorem, 94
7.8 Strict inequalities, 94
7.9 Complementary slackness, 95
7.10 Application: max-flow min-cut, 96
8 The structure of polyhedra 99
8.1 Implicit equalities and redundant constraints, 99
8.2 Characteristic cone, lineality space, affine hull, dimension, 100
8.3 Faces, 101
8.4 Facets, 101
8.5 Minimal faces and vertices, 104
8.6 The face-lattice, 104
8.7 Edges and extremal rays, 105
8.8 Extremal rays of cones, 105
8.9 Decomposition of polyhedra, 106
8.10 Application: doubly stochastic matrices, 107
8.11 Application: the matching polytope, 109
9 Polarity, and blocking and anti-blocking polyhedra 112
9.1 Polarity, 112
9.2 Blocking polyhedra, 113
9.3 Anti-blocking polyhedra, 116
Contents ix
10 Sizes and the theoretical complexity of linear inequalities
and linear programming 120
10.1 Sizes and good characterizations, 120
10.2 Vertex and facet complexity, 121
10.3 Polynomial equivalence of linear inequalities and linear programming, 124
10.4 Sensitivity analysis, 125
11 The simplex method 129
11.1 The simplex method, 129
11.2 The simplex method in tableau form, 132
11.3 Pivot selection, cycling, and complexity, 137
11.4 The worst-case behaviour of the simplex method, 139
11.5 The average running time of the simplex method, 142
11.6 The revised simplex method, 147
11.7 The dual simplex method, 148
12 Primal-dual, elimination, and relaxation methods 151
12.1 The primal-dual method, 151
12.2 The Fourier-Motzkin elimination method, 155
12.3 The relaxation method, 157
13 Khachiyan's method for linear programming 163
13.1 Ellipsoids, 163
13.2 Khachiyan's method: outline, 165
13.3 Two approximation lemmas, 166
13.4 Khachiyan's method more precisely, 168
13.5 The practical complexity of Khachiyan's method, 170
13.6 Further remarks, 171
14 The ellipsoid method for polyhedra more generally 172
14.1 Finding a solution with a separation algorithm, 172
14.2 Equivalence of separation and optimization, 177
14.3 Further implications, 183
15 Further polynomiality results in linear programming 190
15.1 Karmarkar's polynomial-time algorithm for linear programming, 190
15.2 Strongly polynomial algorithms, 194
15.3 Megiddo's linear-time LP-algorithm in fixed dimension, 199
15.4 Shallow cuts and rounding of polytopes, 205
Notes on polyhedra, linear inequalities, and linear
programming 209
Historical notes, 209
Further notes on polyhedra, linear inequalities, and linear programming, 223