Table Of ContentInteger Programming
Mathematics and Its Applications (East European Series)
Managing Editor:
M. FU\ZEVVINJCEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. BIALYNICKI-BIRULA, Institute of Mathematics, Warsaw University, Poland
H. KURKE, Humboldt University Berlin, G.D.R.
J. KURZWEIL, Mathematics Institute, Academy of Sciences, Prague, Czechoslovakia
L. LEINDLER, Bolyai Institute, Szeged, Hungary
L. LOYA sz, Bolyai Institute, Szeged, Hungary
D. S. MITRINOVIC, University of Belgrade, Yugoslavia
S. ROLEWICZ, Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Bl. H. SENDOV, Bulgarian Academy of Sciences, Sofia, Bulgaria
I. T. TODOROV, Bulgarian Academy of Sciences, Sofia, Bulgaria
H. TRIEBEL, University of Jena, G.D.R.
Volume 46
Stanislaw Walukiewicz
System Research Institute,
Polish Academy of Sciences, Warszawa
Integer Progran1n1ing
'~ ·
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Walukiewicz, Stanislaw.
Integer programming ! Stanislaw Walukiewicz.
p. cm. - (Mathematics and its applications I East European series: v. 46)
"Revised translation from the Polish original Programowanie dyskretne, published in 1986"
T.p. verso.
ISBN 978-90-481-4068-8 ISBN 978-94-015-7945-2 (eBook)
DOI 10.1007/978-94-015-7945-2
1. Integer programming. 1. Walukiewicz, Stanislaw.
Programowanie dyskretne. II. Title. III. Series: Mathematics and its applications
(Springer-Science+Business Media, B.V.)
T57.74.W37 1990
519.7'7-dc20
90-4332
ISBN 978-90-481-4068-8 CIP
Revised translation from the Polish original Programowanie dyskretne, published in 1986 by
Pafistwowe Wydawnictwo Naukowe, Warszawa (translated by the Author)
AII Rights Reserved
©
Copyright 1991 by Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1991
Softcover reprint ofthe hardcover Ist edition 1991
No part of the material protected by this copyright notice may be reproduced or utilized in any
form or by any means, electronic or mechanical, inc1uding photocopying, recording, or by any in
formation storage and retrieval system, without written permission from the copyright owner.
Printed in Poland by D.R.P.
To My Mother
Bronislawa Walukiewicz
SERIES EDITOR'S PREFACE
'Et moi, ... , so j'avait su comment en revenir, One service mathematics has rendered the
je n'y serais point al!e.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non
The series is divergent; therefore we may be sense'.
able to do something with it. Eric T. Bell
0. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both
feedback and nonlinearities abound. Similarly, all kinds of parts of mathematics
serve as tools for other parts and for other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such
statements as: 'One service topology has rendered mathematical physics ... '; 'One
service logic has rendered computer science .. .'; 'One service category theory has
rendered mathematics ... '. All armably true. And all statements obtainable this way
form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one
hundred volumes have appeared it seems opportune to reexamine its scope. At the
time I wrote
"Growing specialization and diversification have brought a host of
monographs and textbooks on increasingly specialized topics. However,
the 'tree' of knowledge of mathematics and related fields does not grow
only by putting forth new branches. It also happens, quite often in fact,
that branches which were thought to be completely disparate are suddenly
seen to be related. Further, the kind and level of sophistication of mathe
matics applied in various sciences has changed drastically in recent
years: measure theory is used (non-trivially) in regional and theoretical
economics; algebraic geometry interacts with physics; the Minkowski lem
ma, coding theory and the structure of water meet one another in packing
and covering theory; quantum fields, crystal defects and mathematical
programming profit from homotopy theory; Lie algebras are relevant
to filtering; and prediction and electrical engineering can use Stein
spaces. And in addition to this there are such new emerging subdisciplines
as 'experimental mathematics', 'CFD', 'completely integrable systems',
viii Series Editor's Preface
'chaos, synergetics and large-scale order', which are almost impossible to
fit into the existing classification schemes. They draw upon widely differ
ent sections of mathematics."
By and large, all this still applied today. It is still true that at first sight mathematics
seems rather fragmented and that to find, see, and exploit the deeper underlying
interrelations more effort is needed and so are books that can help mathematicians
and scientists do so. Accordingly MIA will continue to try to make such books
available.
If anything, the description I gave in 1977 is now an understatement. To the
examples of interaction areas one should add string theory where Riemann surfaces,
algebraic geometry, modular functions, knots, quantum field theory, Kac-Moody
algebras, monstrous moonshine (and more) all come together. And to the examples
of things which can be usefully applied let me add the topic 'finite geometry'; a
combination of words which sounds like it might not even exist, let alone be appli
cable. And yet it is being applied: to statictics via designs, to radar /sonar detection
arrays (via finite projective planes), and to bus connections of VLSI chips (via
difference sets). There seems to be no part of (so-called pure) mathematics that is
not in immediate danger of being applied. And, accordingly, the applied mathema
tician needs to be aware of much more. Besides analysis and numerics, the tradi
tional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear
world and the extra mathematical sophistication that this requires. For that is
where is where the rewards are. Linear models are honest and a bit sad and depres
sing: proportional efforts and results. It is in the nonlinear world that infinitesimal
inputs may result in macroscopic outputs (or vice versa). To appreciate what I
am hinting at: if electronics were linear, we would have no fun with transistors and
computers; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis,
superspace and anticommuting integration, p-adic and ultrametric space. All three
have applications in both electrical engineering and physics. Once, complex num
bers were equally outlandish, but they frequently proved the shortest path between
'real' results. Similarly, the first two topics named have already provided a number
of 'wormhole' paths. There is no telling where all this is leading-fortunately.
Thus the original scope of the series, which for various (sound) reasons now
comprises five subseries: white (Japan), yellow (China), red (USSR), blue (Eastern
Europe), and green (everything else), still applies. It has been enlarged a bit to
include books treating of the tools from one subdiscipline which are used in others.
Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical
andjor scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour
into another;
Series Editor's Preface ix
- influences which the results, problems and concepts of one field of enquiry have,
and have had, on the development of another.
Euler wrote
"For since the fabric of the universe is most perfect and the work of a most
wise Creator, nothing at all takes place in the Universe in which some rule
of maximum or minimum does not appear."
That sort of statement implies that there is nothing in mathematics, at leas
applied mathematics, that has not to do with optimization, i.e. programming. Add
lo that that much in this world (possibly all) is discrete and the place of (nonlinear
and linear) integer programming seems assured in the general scheme of things.
And so, of course, it is. Correspondingly, to being in principle applicable to every
thing, even linear integer programming, the subject of this book is very difficult
.¥&'-complete in the language of comutational complexity).
It is also a field in which a great deal is happening at the moment, and that seems
likely to continue, and, finally, it seems to me be an area of research that is less well
known than it should be to fellow mathematical specialists, and it is perhaps insu
fficiently appreciated in terms of the depth and beauty of its problems and results.
(This may well hold for all of discrete mathematics, and, to a lesser extent, for al
gebra as well.)
Thus, I am happy to welcome this volume, suitable both for self-study and for
courses, on the topic of linear integer programming.
The shortest path between two truths in the Never lend books, for no one ever returns
real domain passes through the complex them; the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas seulement The function of an expert is not to be more
!'occasion de resoudre des problemes ... elle right then other people, but to be wrong for
nous fait pressentir Ia solution. more sophisticated reasons.
H. Poincare David Butler
Bussum, September 1989 Michie! Hazewinkel
Preface
Integer programming originated in the mid-fifties as a branch of linear programming.
Now it is a rapidly developing subject in itself, having many common points with
operations research on one hand and with discrete mathematics on the other.
A measurement of this development may be the increasing numbers of publications,
reflected in the classified bibliographies edited by Kasting in 1976 (4704 items),
Hausman in 1978 (3162 items) and by Randow in 1981 (3924 items). The number
of applications of integer programming has also sharply increased in recent years.
The aim of this book is to present in a unified way the theory and methodology
of integer programming including recent results. It contains examples how these
results may be used in the construction of more efficient numerical methods. It is
designed as a textbook for a one- or two-semester course at a department of oper
ations research, management science or industrial administration. It may also be
used as source of references.
In contrast to other books on integer programming, we consistently use here the
notion of equivalence and relaxation: both are defined in Chapter 1. This makes
all our considerations shorter and allows us to introduce simple notations.
The book has ten chapters. Each chapter is to a certain extent a closed unit,
ended, with the exception of Chapter 10, with bibliographical notes and exercises.
The chapters are divided into sections, some longer sections are divided into sub
sections. We use typical numbering of chapters, sections, figures and tables.
The main objective of integer programming is the construction of numerical
methods for solving discrete optimization problems. First we describe the idea of
a method, next discuss results on which it is based, and at the end we present a for
mal description of it in terms of a simple model language similar to ALGOL 60.
Mter such a presentation it would not be difficult to write a computer program for
the considered methods.
The book is organized in the following way.
In Chapter I, we discuss the concept of and give formal definitions for an equiv
alence and relaxation. Next we present examples of applications of integer program
ming, describe ideas of different integer programming methods and of different
ways of measuring their efficiency. In this chapter, we also introduce the notations
which we will use further on.
For the sake of completeness, we review the main results of linear programming
in Chapter 2. New here is a description of the ellipsoid method and discussion
of subgradient methods used increasingly more often in integer programming.
xii Preface
In Chapter 3, we discuss some integer programming problems which may be
solved as linear programming problems and give a description of cutting-plane
methods.
The next four chapters are devoted to branch-and-bound methods as the most
efficient approach to solving integer programming problems. In Chapter 4, we de
scribe the idea of such an approach and the use of linear programming in it, while
in Chapter 7, we consider other relaxations. The second part of Chapter 7 is devoted
to duality in integer programming. Computational experiments show that it is worth
reformulating a given integer programming problem before solving it. Different
ways of reformulating a given problem are considered in Chapter 6. They are mainly
based on the result of analysis of the knapsack problem done in Chapter 5. In Chap
ter 10, we give a general description of a system of computer codes in which the
above-mentioned reformulation plays a very important role and in fact determines
the efficiency of the system.
In Chapter 8, we consider problems with binary matrices. For such particular
problems, similarly as for the knapsack problem, many results have been obtained
and some of them are used in the construction of solution methods for general in
teger programming problems.
Chapter 9 is devoted to near-optimal methods and, in particular, to ones in
which it is possible to estimate the error of such methods.
The book is an extension of my lectures given at the University of Copenhagen
and at Warsaw University. Some parts of it were presented at seminars of the Math
ematical Programming Department (MPD) of the System Research Institute. I wish
to thank my friends from the MPD for their constructive criticism and in particular
Dr. lgnacy Kaliszewski, who read the manuscript and proposed many improve
ments.
Stanislaw Walukiewicz
Description:'Et moi, ... , so j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point al!e.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we
