Integer and Combinatorial Optimization

772 Pages·1988·78.75 MB·English

by  Laurence A. Wolsey

Integer and Combinatorial Optimization
WILEY-INTERSCIENCE
SERIES IN DISCRETE MATHEMATICS AND OPTIMIZATION
ADVISORY EDITORS
RONALD L. GRAHAM
AT & T Laboratories. Florham Park. New Jersey, US.A.
JAN KAREL LENSTRA
Department ofM athematics and Computer Science,
Eindhoven University ofT echnology, Eindhoven, The Netherlands
ROBERT E. TARJAN
Princeton University, New Jersey, and
NEC Research Institute. Princeton, New Jersey. US.A.
A complete list of titles in this series appears at the end of this volume.
Integer and Combinatorial Optimization
GEORGE NEMHAUSER
School of Industrial
and Systems Engineering
Georgia Institute of Technology
Atlanta, Georgia
LAURENCE WOLSEY
Center for Operations Research and Econometrics
Universite Catholique de Louvain
Louvain-Ia-Neuve, Belgium
A Wiley-Interscience Publication
JOHN WILEY & SONS, INC.
New York • Chichester • Weinheim • Brisbane • Singapore • Toronto

To our parents
Preface
The explosion of new results in integer and combinatorial optimization that began about
fifteen years ago inspired us to write a book that would unify theory and algorithms and
could serve as a graduate text and reference for researchers and practitioners. We have
been very excited about many of the new developments that have made it possible to solve
large-scale integer programming problems and that have opened up new areas of research
which surely will yield more robust and efficient algorithms. Little did we realize the
enormity of the task. Both of us worked steadily on this project for more than four years.
The end result was a manuscript of nearly 1400 typewritten pages which, although it does
not come close to covering all of the literature, covers those topics that we believe
constitute the most significant theoretical and algorithmic developments.
Optimization means to maximize (or minimize) a function of many variables subject to
constraints. The distinguishing feature of discrete, combinatorial, or integer optimization
is that some of the variables are required to belong to a discrete set, typically a subset of
integers. These discrete restrictions allow the mathematical representation of phenomena
or alternatives where indivisibility is required or where there is not a continuum of
alternati ves.
Discrete optimization problems abound in everyday life. An important and widespread
area of applications concerns the management and efficient use of scarce resources to
increase productivity. These applications include operational problems such as the distri
bution of goods, production scheduling, and machine sequencing. They also include
planning problems such as capital budgeting, facility location and portfolio selection, and
design problems such as telecommunication and transportation network design, VLSI
circuit design and the design of automated production systems. Discrete optimization
problems also arise in statistics (data analysis), physics (determination of minimum
energy states), cryptography (designing unbreakable codes), politics (selecting fair election
districts), and mathematics (as a powerful technique for proving combinatorial theorems).
Moreover, applications of discrete optimization are in a period of rapid development
because of the widespread use of microcomputers and the data provided by information
systems. This is particularly relevant in the manufacturing sector of the economy where
increased competition and flexibility provided by new technology make it imperative to
seek better solutions from larger and more complex sets of alternatives.
This book is about the mathematics of discrete optimization, which includes the
representation of problems by mathematical models and, especially, the solution of the
models. The focus is on understanding the mathematical underpinnings of the algorithms
that make it possible to solve (exactly or approximately) the large and complex models
that arise in practical applications.
Chapter I.l discusses problem formulation, which is important not only to demonstrate
the scope of applications, but also because the structure of the formulation is of crucial
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viii Preface
importance to solving the model. Chapter 1.1 gives a comprehensive treatment of this
subject.
The remainder of Part I presents mathematics and algorithms that are the foundations
for the discrete optimization theory and techniques of Parts II and III. There are chapters
on well-established subjects including linear programming (Chapter 1.2), graphs and
networks (Chapter 1.3), and computational complexity (Chapter 1.5). The presentation of
polyhedral theory (Chapter 1.4) begins with basic results from linear algebra and then
emphasizes precisely those results that are essential to a fundamental understanding of the
algebra and geometry of the convex hull of a discrete set. Chapter 1.6 gives new algorithms
and results on linear programming and, in particular, establishes the fundamental
connection between separation and optimization. Chapter I. 7 presents a modern treat
ment of the classical problem of solving linear equations in integers and also includes an
introduction to the recent work on reduced bases for integer lattices.
Parts II and III present basic approaches and algorithms for solving discrete optimiza
tion problems. Part II deals with general problems and those that contain some structure.
These are the problems that are hard to solve but, for the most part, they are the ones that
arise in practical applications.
Chapters 11.1 and 11.2 treat the problem of describing the set of feasible solutions to an
integer program by a set of linear inequalities. It begins with elementary ideas, but also
includes a thorough development of advanced topics such as superadditive valid inequali
ties and the use of structure to obtain facet-defining inequalities. Objective functions for
integer programs are introduced in Chapter 11.3 where the fundamental approaches of
relaxation and duality are developed for the purpose of obtaining upper bounds on the
optimal value. Most of the advanced material in these chapters has appeared only in
research articles and monographs, but is essential for the development of future genera
tion algorithms for solving integer programs.
Algorithms are presented in Chapters 11.4, 11.5 and 11.6. Chapter 11.4 presents classical
branch-and-bound and cutting plane algorithms. Specialized algorithms that use varying
degrees of structure to obtain exact or approximate solutions are presented in
Chapters 11.5 and 11.6. Here we study and illustrate a number of techniques that, for the
most part, have been developed over the last decade and are not covered in the currently
available textbooks. These include strong cutting plane algorithms, primal and dual
heuristic analysis, decomposition and reduced bases, and their applications to 0-1 integer
programs, the traveling salesman problem and fixed-charge network flow problems.
Part III treats highly structured combinatorial optimization problems for which elegant
results are known. Chapter 111.1 studies polyhedra with integral extreme points. It includes
classical results on total unimodularity and recent results on totally balanced, balanced,
and perfect matrices and on the blocking and antiblocking theory of polyhedra. Chapters
111.2 and 111.3 are on the classical combinatorial problems of matching and matroids,
respectively. In both of these chapters the emphasis is on optimization algorithms,
polyhedral combinatorics and duality. Chapter 111.3 also introduces the significant role of
submodular and supermodular functions in combinatorial optimization.
Notes appear at the end of each chapter. Their purpose is to reference our source
materials, and to comment briefly on extensions and related topics that are not discussed
in the body of the text. The citations and references are selective. With the exception of
Chapter 1.1, in Part I our objective is to provide foundation material, and thus the notes
are limited to a small number of references that cover the corresponding topics in much
greater detail than is done here. However, in Parts II and III we have attempted to cite the
original papers in which the material appears as well as some other influential works.
The book can be used as a graduate text or for self-guided reading in several ways. Since
we cannot imagine a reader who would want to undertake a straight cover-to-cover
Preface ix
reading and since our experience has shown that it is not possible to cover the whole book
in even a two-semester, graduate level course, it is necessary to be selective in a first
reading.
For graduate students in mathematical programming, especially those planning to
undertake research in discrete optimization, we suggest a full academic year course
(course AY). Three one-semester options are: a course emphasizing practical algorithms
(course PA), a course emphasizing general theory (course GT), and a course in polyhedral
combinatorics (course PC).
Each course should begin with some exposure to Chapter 1.1 on model formulation,
which is important not only to demonstrate the scope of applications, but also because the
structure of the formulation is of crucial importance in solving the model.
Chapters 1.2 and 1.3 are only for review, since it is wise for any reader of the book to have
studied linear programming as a prerequisite. But a typical linear programming course,
unfortunately, does not cover polyhedral theory. Therefore, all courses should cover
Chapter 104. In course PA, just enough of the first four sections should be covered (without
proofs) for the student to understand the concept of facets of polyhedra and the idea of
Theorem 6.1 on the convex hull of a discrete set of points.
The coverage of Chapter 1.5 on computational complexity will depend on the students'
backgrounds and the instructor's taste; but at the very least, students in all courses should
be introduced to the concepts of polynomial computation and NP-completeness. Simi
larly, students in all courses should be introduced to the concept of separation and the
polynomial equivalence of separation and optimization (Section 1.6.3). This should be
done very informally in course PA. Sections 1.6.2, 1.604 and 1.6.5 are independent reading
and should be omitted in a first reading of the book. Chapter 1.7, and then Section 11.6.5,
might be covered only in courses AY and GT ift ime permits at the end of the course. They
can also be omitted in a first reading.
Courses PA and GT focus on different parts of Part II. Course PC can omit Part II
altogether, but would be more interesting if Sections 11.1.1, 11.1.2 (first-half), 11.2.1, 11.2.3,
and 1I.6.3 also were included.
The following sections from Part II are common to courses AY, PA, and GT: 11.1.1,
11.2.1, 11.2.2, 1I.3.l, 11.3.6, 11.3.7, 1104.1, 11.5.1, 11.5.2 and 11.5.3.
Course PA should also cover Sections 1104.2, 11.504, 11.5.5, 11.6.1 (knapsack problem)
and 11.6.2, and, if time permits, Sections 11.204 and 11.6.4. The instructor may find some
time for the important class of problems and algorithms discussed in the later two sections
by omitting or only sketching some proofs from the earlier sections.
Course GT should also cover Sections 11.1.2 (leaving out the subsection on bounded
integer variables), 11.1.3, 11.1.4, 11.1.5, 11.2.3, 11.3.2, and 11.3.3, and the first two sections of
Chapter 111.1. If time permits, additional theoretical material could be selected from
Sections 11.1.6, 11.1.7, 11.304, 11.3.5, or some algorithms could be studied from Sections
11.4.3,11.5.4 and the first three sections of Chapter 11.6.
With respect to Part II, course AY is the union of the material covered in courses PC
and GT. From Part III, course AY should also cover sections 111.1.1, 111.1.2, 111.1.4 and the
first three sections of Chapter 111.2. Any remaining time could be spent on either
sections 111.1.5, 111.1.6, or the first few sections of Chapter 111.3.
The material to be selected from Part III for course PC can vary according to taste. We
suggest all of Chapter 111.1 except for Section 111.1.3, the first 3 sections of Chapter 111.2
and the first 5 sections of Chapter 111.3.
This book could not have been written without the tremendous support that we
received from the Center for Operations Research and Econometrics (CORE) of the
Universite Catholique de Louvain, and thus we are extremely grateful to Jacques Dreze,
the founder and intellectual leader of CORE.
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We met at CORE in the winter of 1970. Nelnhauser spent the academic year 1969-1970
at CORE and Wolsey presented a seminar on his early work on the group-theoretic
approach to integer programming. Subsequently, Wolsey became a permanent member of
CORE and Nemhauser returned to CORE for the period 1975-1977 as Research
Director. During this period, the authors collaborated extensively on research in the
analysis of heuristics and other topics in integer and combinatorial optimization stimu
lated by an active research group that included Jack Edmonds, Bob Bland, Guy de
Ghellinck, Rick Giles, Bob Jeroslow, Tom Magnanti, Bill Pulleyblank, Mike Ball and
Gerard Cornuejols. All of these people, as well as our Dutch neighbors, Jan-Karel Lenstra
and Alexander Rinnooy Kan, contributed to our understanding of the subject and
motivation to write a book.
A NATO research grant made it possible for us to continue our research collaboration
through the late seventies and early eighties, and we began to draft the manuscript
earnestly during Nemhauser's fourth year at CORE in 1983-1984. During the writing of
the book we benefitted from numerous discussions with our friends and professional
colleagues including Egon Balas, Vasek Chvatal, Marshall Fisher, Martin Grotschel, Ellis
Johnson, Manfred Padberg. Lex Schrijver, Jorgen Tind and Les Trotter. We are particu
larly grateful to Gerard Cornuejols who read Parts I and II and provided extensive
comments and suggestions and to Bill Pulleyblank who did the same for Part III. We are
also thankful for the comments we received on various drafts of the text from Jorgen Tind,
Bob Jeroslow, Alan Goldman, Anton Kolen, Jan-Karel Lenstra, Lex Schrijver, Donna
Crystal Llewellyn, Martin Dyer, Mike Todd, Jean-Philippe Vial and John Vande Vate.
Our students in courses given at CORE, Cornell and Georgia Tech found typos and other
mistakes that otherwise would have been missed~ special thanks are due to Ronny Aboudi,
Yves Pochet and Gabriele Sigismondi.
The chores of deciphering our untidy handwritten drafts and of retyping endless
revisions were done graciously and with utmost care and patience by the late Elizabeth
Pecquereau, formerly a secretary at CORE. We are very sad that we will not be able to
share the joy of seeing the final product with our dear friend Elizabeth. Fabienne Henry of
CORE and Yvonne Kissi of Georgia Tech also did excellent jobs in typing parts of the
manuscript. Sheila Verkaren of CORE always managed to spare some of Elizabeth's or
Fabienne's time for our book, even though we were using far more than our fair share of
CORE's secretarial resources.
Over a period of four years, Ellen Nemhauser and Marguerite Wolsey were frequently
ignored while their husbands spent evenings and weekends writing, and occasionally were
imposed upon by a boarder who ate and slept at their house, but otherwise was too
involved in mathematics to engage in civil conversation or to wash the dishes. We thank
them for their love and patience and hope to make amends.
GEORGE L. NEMHAUSER
LAURENCE A. WOLSEY
Allan/a, Georgia, USA
Louvain-Ia-Neuve, Belgium
February, 1988.
Contents
PART I. FOUNDATIONS 1
1.1 The Scope of Integer and Combinatorial Optimization 3
1. Introduction 3
2. Modeling with Binary Variables I: Knapsack, Assignment and Matching,
Covering, Packing and Partitioning 5
3. Modeling with Binary Variables II: Facility Location, Fixed-Charge
Network Flow, and Traveling Salesman 7
4. Modeling with Binary Variables III: Nonlinear Functions and
Disjunctive Constraints 10
5. Choices in Model Formulation 14
6. Preprocessing 17
7. Notes 20
8. Exercises 22
1.2 Linear Programming 27
1. Introduction 27
2. Duality 28
3. The Primal and Dual Simplex Algorithms 30
4. Subgradient Optimization 41
5. Notes 49
1.3 Graphs and Networks 50
1. Introduction 50
2. The Minimum-Weight or Shortest-Path Problem 55
3. The Minimum-Weight Spanning Tree Problem 60
4. The Maximum-Flow and Minimum-Cut Problems 62
5. The Transportation Problem: A Primal-Dual Algorithm 68
6. A Primal Simplex Algorithm for Network Flow Problems 76
7. Notes 82
1.4 Polyhedral Theory 83
1. Introduction and Elementary Linear Algebra 83
2. Definitions of Polyhedra and Dimension 85
3. Describing Polyhedra by Facets 88
4. Describing Polyhedra by Extreme Points and Extreme Rays 92
5. Polarity 98
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