Table Of ContentNetwork Optimization
V. K. Balakrishnan
Department of Mathematics
University of Maine
Orono, Maine
USA
CRC Press
Taylor & Francis Croup
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Croup, an informa business
A CHAPMAN & HALL BOOK
Opening verse in Kalidasa's Raghuvamsam
(Salutations to Parvathi and Parameswara - the Mother and the Father of the
Universe-who are connected to each other the way a word and its meaning
are linked for the complete understanding of the meaning of a word.)
Dedicated to the fond memory of
my parents
who convinced me that a quest for knowledge
is the most worthwhile of all our pursuits
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© 1995 V. K. Balakrishnan
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Contents
Preface vii
1 Trees, arborescences and branchings 1
1.1 Some graph theory concepts 1
1.2 Spanning trees 5
1.3 Minimum weight spanning trees 10
1.4 The traveling salesman problem 19
1.5 Minimum weight arborescences 24
1.6 Maximum weight branchings 29
1.7 Exercises 37
2 Transshipment problems 40
2.1 The network simplex method 40
2.2 Initialization, feasibility and decomposition 47
2.3 Inequality constraints 54
2.4 Transportation problems 56
2.5 Some applications to combinatorics 61
2.6 Assignment problems 74
2.7 Exercises 82
3 Shortest path problems 89
3.1 Some shortest path algorithms 89
3.2 Branch and bound methods for solving the traveling
salesman problem 98
3.3 Medians and centers 111
3.4 The Steiner tree problem 112
3.5 Exercises 117
4 Minimum cost flow problems 122
4.1 The upper-bounded transshipment problem 122
4.2 Initialization and feasibility 126
vi Contents
4.3 The maximum flow problem 131
4.4 The MPM algorithm 141
4.5 The MFP in planar undirected networks 151
4.6 Multiterminal maximum flows 155
4.7 The minimum cost flow problem 162
4.8 Flow problems and combinatorics 171
4.9 Exercises 184
5 Matchings in graphs 194
5.1 Cardinality matching problems 194
5.2 The stable marriage problem 212
5.3 Weighted matching problems 215
5.4 The Chinese postman problem 224
5.5 Exercises 228
Further reading 232
References 233
Solutions to selected problems 235
Index 245
Preface
Problems in network optimization arise in all areas of technology and
industrial management. The topic of network flows, in particular, has applica-
tions in such diverse fields as chemistry, communications, computer science,
economics, engineering, management science, scheduling and trans- portation,
to name but a few. The aim of this book is to present the important concepts of
network optimization in a concise textbook suitable for upper- level under-
graduate students in computer science, mathematics and opera- tions research.
While discussing algorithms, the emphasis in this book is more on clarity and
plausibility than on complexity considerations. At the same time, rigorous
arguments have been used in proving the basic theorems. A course in
introductory linear algebra is the only prerequisite needed to follow the
material presented in the text.
The basic graph theory concepts are presented in Chapter 1, along with
problems related to spanning trees and branching. Network flow problems are
then treated in detail in Chapters 2-4. It is the network simplex algorithm that
holds these three chapters together. The theory of matching in both bipartite
and nonbipartite graphs is covered in Chapter 5. The relation between network
flow problems and certain deep theorems in combinatorics - a topic usually not
covered in undergraduate texts - is established in detail in sections 2.6 and
4.8, culminating in a grand equivalence theorem as displayed in Figure 4.10.
Each chapter ends with several exercises, ranging from routine numerical
ones to challenging theoretical problems, and a student interested in learning
the subject is expected to solve as many of these as possible.
This book is based on topics selected from courses on graph theory, linear
programming, discrete mathematics, combinatorics and combinatorial opti-
mization taught by me to undergraduate and beginning graduate students in
computer science, electrical engineering, forest management and mathema-
tics at the University of Maine during the last decade. The contribution of my
students is implicit on every page of this text. It is indeed a pleasure to
acknowledge their participation and excitement in this collective learning
process.
viii Preface
My indebtedness in writing this monograph also encompasses many
sources, including the tomes mentioned in the reading list and in the list of
references at the end of the book and several mathematicians with whom
I have come into contact in the last few years at national and international
conferences and workshops. In particular, I would like to thank Vasek
Chvatal, Victor Klee, George Nemhauser, James Orlin, Christos Papadimit-
riou, Robert Tarjan and Herbert Wilf-creative mathematicians who also
happen to be truly inspiring teachers. Eugene Lawler (who is no longer with
us) was always a source of inspiration and guidance for the completion of this
project, and I am beholden to him for ever. He will be dearly missed. Special
thanks also go to the reviewer from Chapman & Hall who patiently read the
manuscript more than once and offered many constructive suggestions and
helpful comments. If there are still errors or omissions, the responsibility is
entirely mine.
I am grateful to my colleagues and the administrators at the University of
Maine for providing me with necessary facilities and opportunities to
undertake and finalize this project.
In conclusion, I would like to express my profound gratitude to the
editorial and production staff at Chapman & Hall, particularly to my
commissioning editor, Achi Dosanjh, for her unfailing cooperation and
continual support during the entire reviewing and editing process of this
book, and to my sub-editor, Howard Davies, for taking care of several
last-minute corrections along with his multifarious sub-editing responsibil-
ities. I also thank with great pleasure my initial commissioning editor, Nicki
Dennis, for the strong encouragement she gave me while accepting my
proposal. Finally, I would like to add that I owe a great deal to the keen
interest and abiding affection my family has shown me at every stage of this
work.
V. K. Balakrishnan
University of Maine
January 1995
1
Trees, arborescences
and branchings
1.1 SOME GRAPH THEORY CONCEPTS
Graphs and digraphs
A graph G = (V,E) consists of a finite nonempty set V and a collection £ of
unordered pairs from V (i.e. two-element subsets of V). Every element in V is
called a vertex of the graph, and each unordered pair in £ is called an edge of the
graph. The edge e = {x, y] is an edge between the two vertices x and y which is
incident to both x and y. Two vertices are adjacent to each other if there is an
edge between them: the edge in that case is said to join the two vertices. Two or
more edges that join the same pair of vertices are called parallel edges. A graph
without any parallel edges is a simple graph, in which case the collection £ is
a set. Otherwise G is multigraph. A graph G' is a subgraph of the graph G if every
vertex of G' is a vertex of G and every edge of G' is an edge of G.
A directed graph or digraph consists of a finite set V of vertices and
a collection A of ordered pairs from V called the arcs of the digraph. The
digraph is a simple digraph if A is a set. If a = (x, y) is an arc, then a is an arc
from the vertex x to the vertex y and is incident (adjacent) from x and incident
(adjacent) to y. Two vertices are nonadjacent if there is no arc from one to the
other. The underlying graph G of a digraph D is the graph G obtained from D by
replacing each arc (x, y) by an edge {x, y}.
In this book, unless otherwise mentioned, all graphs are simple graphs and
all digraphs are simple digraphs.
If we associate one or more real numbers with each edge (or arc) of a graph (or
a digraph), the resulting structure is known as a weighted graph or a network.
Connectivity
A path between vertex x and vertex x in a graph is a sequence X ,e ,x ,e ,
t r 1 1 2 2
x ,e ,...,e _ ,x where X ,x ,x ,... are vertices and e is the edge between
3 3 r 1 r 1 2 3 k
