Graphs, Networks and Algorithms (Algorithms and Computation in Mathematics)

655 Pages·2007·5.57 MB·English

by  Dieter Jungnickel

Algorithms and Computation
in Mathematics Volume 5
•
Editors
ArjehM.Cohen HenriCohen
DavidEisenbud MichaelF.Singer BerndSturmfels
Dieter Jungnickel
Graphs, Networks
and Algorithms
Third Edition
With209Figuresand9Tables
ABC
Dieter Jungnickel
Universität Augsburg
Institut für Mathematik
86135Augsburg
Germany
E-mail:[email protected]
LibraryofCongressControlNumber:2007929271
MathematicsSubjectClassification(2000):11-01,11Y40,11E12,11R29
ISSN1431-1550
ISBN 978-3-540-72779-8SpringerBerlinHeidelbergNewYork
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A mathematician, like a painter or a poet,
is a maker of patterns.
If his patterns are more permanent than theirs,
it is because they are made with ideas.
G. H. Hardy
To my teacher, Prof. Hanfried Lenz
Preface to the Third Edition
The show must go on.
Ira Gershwin
This new third edition has again been thoroughly revised, even though the
changes are not as extensive as in the second edition. Of course, the general
aims of the book have remained the same.
In particular, I have added some additional material, namely two new
sectionsconcerninggraphicalcodes(whichprovidesalessobvious areaofap-
plication and, as I hope, might also interest the reader in the important field
of coding theory) and about two dozen further exercises (as usual, with so-
lutions). I have also discussed and referenced recent developments, especially
forthetravellingsalesmanproblem,wheretrulyimpressivenewworldrecords
have been achieved. Moreover, the presentation of the material has been im-
proved in quite a few places, most notably in the chapters on shortest paths
and colorings. In addition to this, many smaller changes and corrections have
been made, and the proof of several theorems have been rewritten to make
them more transparent, or more precise.
Again, I thank my students and assistants for their attention and interest
aswellastheinputtheyprovided.Moreover,Iamindebtedtoseveralreaders
whoalertedmetosome(fortunately,moreorlessminor)problems;andIam,
of course, also grateful for the encouraging comments I have received.
Augsburg, May 2007 Dieter Jungnickel
Preface to the Second Edition
Change is inevitable...
Change is constant.
Benjamin Disraeli
When the first printing of this book sold out in a comparatively short time,
it was decided to reprint the original edition with only small modifications:
I just took the opportunity to correct a handful of minor mistakes and to
provideafewupdatestothebibliography.Incontrast,thenewsecondedition
has been thoroughly revised, even though the general aims of the book have
remained the same. In particular, I have added some new material, namely
a chapter on the network simplex algorithm and a section on the five color
theorem; this also necessitated some changes in the previous order of the
presentation (so that the numbering differs from that of the first edition,
beginning with Chapter 8). In addition to this, numerous smaller changes
and corrections have been made and several recent developments have been
discussed and referenced. There are also several new exercises.
Again, I thank my students and assistants for their attention and interest
as well as the input they provided. Moreover, I am particularly grateful to
two colleagues: Prof. Chris Fisher who read the entire manuscript and whose
suggestions led to many improvements in the presentation; and Priv.-Doz.
Dr.BernhardSchmidtwholetmemakeuseofhislecturenotesonthenetwork
simplex algorithm.
Augsburg, September 2004 Dieter Jungnickel
Preface to the First Edition
Thealgorithmicwayoflifeisbest.
Hermann Weyl
During the last few decades, combinatorial optimization and graph theory
have – as the whole field of combinatorics in general – experienced a partic-
ularly fast development. There are various reasons for this fact; one is, for
example, that applying combinatorial arguments has become more and more
common.However,twodevelopmentsontheoutsideofmathematicsmayhave
been more important: First, a lot of problems in combinatorial optimization
arose directly from everyday practice in engineering and management: de-
termining shortest or most reliable paths in traffic or communication net-
works, maximal or compatible flows, or shortest tours; planning connections
in traffic networks; coordinating projects; solving supply and demand prob-
lems. Second, practical instances of those tasks which belong to operations
research have become accessible by the development of more and more effi-
cient computer systems. Furthermore, combinatorial optimization problems
are also important for complexity theory, an area in the common intersec-
tion of mathematics and theoretical computer science which deals with the
analysis of algorithms. Combinatorial optimization is a fascinating part of
mathematics, and a lot of its fascination – at least for me – comes from its
interdisciplinarity and its practical relevance.
The present book focuses mainly on that part of combinatorial optimiza-
tion which can be formulated and treated by graph theoretical methods;
neither the theory of linear programming nor polyhedral combinatorics are
considered. Simultaneously, the book gives an introduction into graph the-
ory,wherewerestrictourselvestofinitegraphs.Wemotivatetheproblemsby
practicalinterpretationswhereverpossible.1 Also,weuseanalgorithmicpoint
of view; that is, we are not content with knowing that an optimal solution
exists (this is trivial to see in most cases anyway), but we are mainly inter-
ested in the problem of how to find an optimal (or at least almost optimal)
1 Most of the subjects we treat here are of great importance for practical appli-
cations, for example for VLSI layout or for designing traffic or communication
networks. We recommend the books [Ber92], [KoLP90], and [Len90].
XII Preface
solution as efficiently as possible. Most of the problems we treat have a good
algorithmic solution, but we also show how even difficult problems can be
treated (for example by approximation algorithms or complete enumeration)
usingaparticularhardproblem(namelythefamoustravelling salesman prob-
lem) as an example. Such techniques are interesting even for problems where
it is possible to find an exact solution because they may decrease the amount
of calculations needed considerably. In order to be able to judge the quality
of algorithms and the degree of difficulty of problems, we introduce the basic
ideas of complexity theory (in an informal way) and explain one of the main
openproblemsofmodernmathematics(namelythequestionP=NP?).Inthe
firstchaptersofthebook,wewillpresentalgorithmsinaratherdetailedman-
ner but turn to a more concise presentation in later parts. We decided not to
include any explicit programs in this book; it should not be too difficult for a
reader who is used to writing programs to transfer the given algorithms. Giv-
ing programs in any fixed programming language would have meant that the
book is likely to be obsolete within a short time; moreover, explicit programs
would have obscured the mathematical background of the algorithms. How-
ever, we use a structured way of presentation for our algorithms, including
special commands based on PASCAL (a rather usual approach). The book
containsalotofexercisesand,intheappendix,thesolutionsorhintsforfind-
ing the solution. As in any other discipline, combinatorial optimization can
be learned best by really working with the material; this is true in particular
for understanding the algorithms. Therefore, we urge the reader to work on
the exercises seriously (and do the mere calculations as well).
Thepresentbookisatranslationofarevisedversionofthethirdeditionof
myGermantextbookGraphen, Netzwerke und Algorithmen.Thetranslation
and the typesetting was done by Dr. Tilla Schade with my collaboration.
The text is based on two courses I gave in the winter term 1984/85 and
in the summer term 1985 at the Justus-Liebig-University in Gießen. As the
first edition of the book which appeared in 1987 was received quite well,
a second edition became necessary in 1990. This second edition was only
slightly changed (there were only a few corrections and some additions made,
including a further appendix and a number of new references), because it ap-
peared a relatively short time after the first edition. The third edition, how-
ever, was completely revised and newly typeset. Besides several corrections
and rearrangements, some larger supplements were added and the references
brought up to date. The lectures and seminars concerning combinatorial op-
timization and graph theory that I continued to give regularly (first at the
University of Gießen, then since the summer term 1993 at the University of
Augsburg)wereveryhelpfulhere.Iusedthetextpresentedhererepeatedly;I
also took it as the basis for a workshop for high school students organized by
the Verein Bildung und Begabung. This workshop showed that the subjects
treated in this book are accessible even to high school students; if motivated
sufficiently, they approach the problems with great interest. Moreover, the
German edition has been used regularly at various other universities.
Preface XIII
I thank my students and assistants and the students who attended the
workshop mentioned above for their constant attention and steady interest.
Thanksaredue,inparticular, toPriv.-Doz.Dr.DirkHachenbergerandProf.
Dr. Alexander Pott who read the entire manuscript of the (German) third
edition with critical accuracy; the remaining errors are my responsibility.
Augsburg, May 1998 Dieter Jungnickel