Table Of ContentCombinatorial
Optimization:
Networks
and Matroids
EUGENE L. LAWLER
University of California at Berkeley
HOLT, RINEHART AND WINSTON
New York Chicago San Francwo Atlanta
Dallas Montreal Toronto London Sydney
To my parents,
and to my children, Stephen and Susan
Copyright 0 1976 by Holt, Rinehart and Wins.on
All rights reserved
Library of Congress Cataloging in Publication Data
Lawler, Eugene L
Combinatorial optimization.
Includes bibliographical references.
1. Mathematical optimization. 2. Matroids.
3. Network analysis (Planning). 4. Computalional
complexity. 5. Algorithms. I. Title.
QA402.5.L39 519.7 76-13516
ISBN 0 -03-084866-o
Printed in the United States of America
8 9 038 9 8 7 6 5 4 3 2
Preface
Combinatorial optimization problems arise everywhere, and certainly in
all areas of technology and industrial management. A growing awareness
of the importance of these problems has been accompanied by a combina-
torial explosion In proposals for their solution.
This, book is concerned with combinatorial optimization problems
which can be formulated in terms of networks and algebraic structures
known as matroids. My objective has been to present a unified and fairly
comprehensive survey 01‘ solution techniques for these problems, with
emphasis on “augmentation” algorithms.
Chapters 3 through 5 comprise the material in one-term courses
on network flow theory currently offered in many university departments
of operations research and industrial engineering. In most cases, a course
in linear programming is designated as a prerequisite. However, this is
not essential. Chapter 2 contains necessary background material on linear
programming, and graph theory as well.
Chapters 6 through 9 are suitable for a second course, presumably
at the graduate level. The instructor may wish to omit certain sections,
depending upon the orientation of the students, as indicated below.
The book is also suitable as a text, or as a reference, for (courses on
combinatorial computing and concrete computational complexity in de-
partments (of computer science and mathematics. Any computer scientist
intending to do serious research on combinatorial algorithrns should have
a working knowledge of the material in this book.
The reader should be aware that certain algorithms are easy to
explain, to understand, and to implement, even though a proof of their
validity may be quite difficult. A good example is the “primal” matroid
intersection algorithm presented in Section 10 of Chapter 8. I can well
imagine situations in which an instructor might legitimately choose to
present this algorithm and its applications, without discussiqg its theoretical
justification in Section 9.
111
Conversely, there are algorithms whose theoretical justification is
not too hard to understand, in principle, but whose detailed implementa-
tion is quite complex. An instructor might well (choose to discuss the
duality theory underlying the weighted non bipartite matching algorithm,
going as far as Section 9 of Chapter 6 and skipping the material in Sections 10
and 11. I might mention, incidentally, that the algorithm in Section 11 is
the one instance in which I have had cause to regret using a simple iterative
description of the algorithms in this book. In this case, a few simple pro-
cedure declarations would have simplified matters substantially.
I began work on this book in the fall of 1968. In my innocence, I
contemplated a two-year project. I did not know that, after many inter-
ruptions, I would still be laboring on the book more than seven years later.
Needless to say, there was much progress in the technical area
during this seven-year period. I managed to make a few contributions
myself, including the development of matroid intersection algorithms and
an O(n3) implementation of Edmonds’ algorithm for weighted nonbipartite
matching. Naturally, these are in the book. There are some results which
I did not put into the book, and possibly should have. These include the
O(n5’*) algorithm for unweighted bipartite matching of Dinic and of
Hopcroft and Karp, and its recent generalization to the nonbipartite case
by Even and Kariv. These must await a second edition, if there is to be one.
Because the writing of this book extended over such a considerable
period, I have had the opportunity to recelive advice, assistance, and en-
couragement from a large number of people. A special word is necessary
for some of them.
Ray Fulkerson was very kind in giving me advice in the early stages
of writing. I am most fortunate to have received his counsel. His untimely
death in January 1976 was a blow to all of us who k.new him, as a scholar
and as a friend.
The last half of this book exists only because of the pioneering insights
of Jack Edmonds. He originated the key ideas for nonbipartite matching,
matroid optimization, and much, much more. I am happy to acknowledge
my personal debt to his creative and fertile mind.
Victor Klee, as consulting editor, was extremely helpful. His crit-
icisms, based on classroom use of the manuscript, were particularly useful
in revising Chapter 4.
Harold Gabow helped me avoid a number of blunders in Chapter 6.
(I, of course, retain sole proprietorship over the errors which remain, as
elsewhere in the book.)
Stein Krogdahl, whose name is seen in many lemmas in Chapter 8,
provided may definitions and proofs where mine were either incorrect or
incomprehensible.
Donald Knuth somehow found time to communicate many useful
suggestions and provided much appreciated encouragement.
The Air Force Office of Scientific Research provided support which
contributed to the writing of this book. I am indebted to that office and to
Captain Richard Rush.
I am also indebted to Nicos Christofides, Stuart Dreyfus, Alan
Frieze, Dan Gussfield, T.C. Hu, Richard Karp, Sukhamay Kundu. Ben
Lageweg, Howard Landeman. Jan Karel Lenstra, Francesco Maffioli,
Colin McDiarmid, George Minty, Katta Murty, Alexander Rinnooy Kan,
Arnon Rosenthal, Phil Spria, John Suraballe, Robert Tarjan, Roger Tobin,
Klaus Truemper, Robert Urquhart, Dominic Welsh, Lee White, and
Norman Zadeh.
The manuscript was typed and retyped by Sharon Bauerle, Doris
Simpson, Ruth Suzuki, and many others. I thank them all.
Rocquencourt, France E.L.L.
July 1976
V
Contents
PYefim 111
Chapter 1
INTRODUCTION I
1. What is Combinatorial Optimization? 1
2. Some Representarive Optimization Problems 2
3. When is a Problem Solved? 4
4. The Criterion qf’Pol~womiu1 Boundedness 5
5. Some Appurentl?) Nonpolynomial-Bounded Problems 8
6. Methods qf’Solutr’on 10
Comments and Rt@rences 12
Chapter 2
MATHEMATICAL PREL.IMINARIES
1.5
1. Muthemuticul Prerequisites 1.5
2. Sets und Relations 16
3. Gruphs and Digraphs 20
4. Subgraphs, Cliques, Multigraphs 24
5. Connectivity in Gruphs 26
6. Connectkitg in Digraphs 28
7. Coc:ycles and Directed Cocycles 31
8. Planarity and Duality 32
9. Euleriun and Hamiltonian Graphs 36
10. Linear Programming Problems 39
Il. The Sim,plex Method 43
12. Geometric Interpretation 48
13. Duality Theory 53
Comments and References 58
vii
Chapter 3
SHORTEST PATHS 59
I. Introduction 59
2. Some Problem Formulations 61
3. Bellman’s Equations 65
4. Acyclic Networks 68
5. Networks with Positive Arcs: Dijksfra’s Method 70
6. Solution by Successive Appro.ximations: Bellman-Ford Method
Method 74
7. Improvements in EfJiciency : Yen ‘s Modifications 76
8. Linear Programming Interpretation and Relaxation
Procedures 77
9. Shortest Paths between all Pairs oj Nodes: .iMatrix
Multiplication 82
10. Floyd- Warshall Method 86
Il. Detection of’ Negative C.ycles 90
12. Networks with Transit Times 92
13. The Minimal Cost-to-time Ratio Cycle Problem 94
14. A4 Shortest Paths: Dreyfus Method 98
15. M Shortest Paths without Re>aeated Nodes 100
Comments and Rejerences 104
Chapter 4
NETWORK FLOWS 109
1. Introduction 109
2. Maximal Flows 110
3. Maximal Flow Algorithm 114
4. Ejficiency ojthe Maximal Fl(ow Algorithm 116
5. Combinatorial Implications of Max-Flow Akin-Cut Theorem 120
6. Linear Programming Interpretation oj Max-Flow Mm-Cut
Theorem 123
7. Minimum Cost Flows 129
c3 Networks with Losses and Gains 134
9. Lower Bounds and Circulations 138
10. The Out-oj:Kilter Method 142
11. Theoretical Improvement in Eficiency ojOOut-qf:Kilter
Method 157
12. Integrality qf’Flows and the iUnimodular Property 160
13. Application to Project Scheduling 16.5
14. Transhipment and Transpormtion Problems 169
Vlll
15. Multitejrminal and Multicommodity Flows 173
Comments and References 177
Chapter 5
BIPARTITE: MATCHING 182
1. Introduiuction 182
2. Problem Reduct&!ms and Equivalences 184
3. Counterparts of’lVetwork Flow Theorems 188
4. Mendelsohn-Dulmage Theorem 191
5. Cardinality Matching Algorithm 193
6. A Special Case: Convex Graphs 196
7. Max-Min Matching 197
8. The Hungarian Methodjor Weighted Matching 201
9. A Special Case: Gilmore-Gomory Matching 207
10. A Novel Optimiz.ation Criterion : Gale-Shapley Matching 211
Comments and References 214
Chapter 6
NONBIPARTITE. MATCHING 217
1. Introduction 217
2. Problem Formuk~tions 218
3. Bidirected Flows 223
4. Augmenting Puths 226
5. Trees and Blossoms 229
6. Cardinulity Matching Algorithm 233
7. Duality Theory 239
8. Linear Programming Formulation of’ Weighted Matching
Problem 242
9. An O(n”) Weighted Matching Algorithm 247
10. An O(n3) Weighted Matching Algorithm 252
11. The Chinese Postman’s Problem 259
Comments and References 261
Chapter 7
MATROIDS AND THE GREEDY ALGORITHM 264
1. Introduction 264
2. Three Apparently Unrelated Optimization Problems 265
3. Matroid Definitions 268
4. Matching, Transversal, and Partition Mutroids 271
iX
5. Mutroid Axiomatics 273
6. The Matroid Greedy Algorithm 275
7. Applications of the Greedy Algorithm 278
8. Matroid Duality 280
9. Variations oj’ the Greedy Algorithm 283
10. Prim Spanning Tree Algorithm 285
11. An Application: Flow Network Synthesis 286
12. The Steiner Problem and Other Dilemmas 290
Comments and Rejerences 296
Chapter 8
MATROID INTERSECTIONS 300
1. Introduction 300
2. Problem Formulations 301
3. Augmenting Sequences and Border Graphs 306
4. Cardinality Intersection Algorithm 313
5. Duality Theory 315
6. Generalized Mendelsohn-Dulmage Theorem, Mutroid Sums
and Matroid Partitions 317
7. Matroid Partitioning Algorithm 320
8. The Shannon Switching Game 324
9. Weighted Augmenting Sequences 326
10. Primal Weighted Intersectiorl Algorithm 332
11. Matroid Polyhedra 334
12. Explanution of’ Primal-Dual .iMethod 339
13. Primal-Dual Weighted Intersection Algorithm 345
14. A Special Case: Directed Spanning Trees 348
Comments and References 352
Chapter 9
THE MATROID PARITY PROBLEM 356
I. Introduction 356
2. Problem Formulations 3.58
3. Augmenting Sequences 363
4. Generalizations 364
Comments and References 367
Author Index 368
Sub jet t Index 371
