Table Of ContentLecture Notes in Economics and Mathematical Systems 300
Ulrich Derigs
Programming
in Networks and
Graphs
On the Combinatorial Background
and Near-Equivalence of Network
Flow and Matching Algorithms
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
300
Ulrich Derigs
Programming in Networks
and Graphs
On the Combinatorial Background
and Near-Equivalence of
Network Flow and Matching Algorithms
Springer-Verlag Berlin Heidelberg GmbH
EditorialBoard
H.Albach M.Beckmann (ManagingEditor) P.Ohrymes
G.Fandel G.Feichtinger J.Green W.Hildenbrand W.Krelle(ManagingEditor)
H.P.Klinzi K.Ritter R.Sato U.Schittko P.Schonfeld R.Selten
ManagingEditors
Prof.Dr.M. Beckmann
Brown University
Providence, RI02912, USA
Prof. Dr.W.Krelle
Institutfi.irGesellschafts-und Wirtschaftswissenschaften
derUniversitatBonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Author
Prof.Dr. Dr.UlrichOerigs
LehrstuhlfUrBetriebswirtschaftslehre, insbesondereBetriebsinformatik
UniversitiitBayreuth,Universitiitsstr. 30, 0·8580Bayreuth,FRG
ISBN 978-3-540-18969-5 ISBN 978-3-642-51713-6 (eBook)
DOI 10.1007/978-3-642-51713-6
libraryofCongressCatalcqinq-in-PublicationData.Derigs,Ulrich, 1950-Programminginnet
worksandgraphs.(Lecture notesineconomicsandmathematicalsystems;300) Bibliography:p.
1.Combinatorialoptimization.2.Networkanalysis(Planning)3.Matchingtheory.I.Title.II.Series.
QA402.5.D47 1988511'.688-3163
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Originallypublished bySpringer-VerlagBerlin HeidelbergNew Yorkin1988
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Meinen Eltern in Dankbarkeit gewidmet
C'est le temps que tu as perdu pour ta rose
qui fait ta rose si importante.
( A. de Sainte Exupery )
Preface
Since the origins of combinatorial optimization network flow-, matching
and certain matroid problems have been treated as cornerstone problems of
this discipline. For each of these problems a variety of algorithms was devel
oped which were able to solve even large-scale, real-world problems. This last
property made these classes outstanding and attractive.
During the seventies with the development of the "complexity theory" of
computer algorithms the uni verse of combinatorial optimization problems was
divided into two major classes: the "easy" problems (Le. those for which a
polynomial algorithm was known) and the "hard" problems (Le. the so-called
NP-complete problems which all become easy if the existence of a polynomial
algorithm for only one representative can be shown).
The last break-through in this direction was done with the development of
the ellipsoid-algorithm for linear programs and the understanding of its impact
for combinatorial optimization by setting up a general ellipsoid-cutting-plane
scheme for combinatorial optimization problems. Yet aIthough by this frame
work a number of special combinatorial problems could newly be classified to be
easy and the borderline between "easy" and "hard" could be discovered almost
completely, this result is not extremely satisfactory since the basic ingredients
of this approach are by no means combinatorially natured - at least the com
binatorial backbones have not been discovered so far. Moreover this scheme
seems not at all to be practical in general.
Thus when looking for nice combinatorial algorithms which work in poly
nomial time one is basically back to the roots, but now knowing for certainty
that
network flow problems
matching problems and
matroid (intersection) problems
are the only major combinatorial optimization problems of practical relevance
which are "easy" under this restriction.
Now there is an immediate interest in analysing and understanding the
special charme or nature of these three problems which makes them "easy".
The construction of a "super-problem" which contains all three problems as
special cases and which itself is also "easy" would certainly help to decode this
property.
Yet so far one did only succeed in constructing easy super-problems for
any two of these three cornerstone-problems - thereby creating another three
corners tone-pro blems:
the matroid matching problem
the submodular flow problem
the general matching problem in bidirected graphs.
In its general form, the matroid-intersection problem is of purely theoretical
interest only and from a practical point of view network flow problems, and since
only recently, matching problems, too, play the dominant role. Both problems
were generalized to the so-called general matching problem or bidirected flow
problem. Yet what has even more consequences is the fact that one could show
that network flow problems can be reduced to a genuine matching problem.
Thus we are really left with two comerstone problems only.
In the literature "network flow" and "matching" are treated separately in
general and for each class a variety of "different" algorithms has been devel
oped up to now. In fact from an algorithmic point of view the development of
matching procedures was influenced significantly by network flow techniques.
Knowing the relation among both problems one might expect a relation
among these procedures, too. The question that we want to contribute to in
VI
this tract is concerned with the existence of a COInmOn combinatorial principle
which might be inherent in all those at first sight different approaches.
In fact we will show that all common network flow and matching algo
rithms follow (implicitely) the "shortest augmenting path" concept which can
be interpreted as a greedy-like decision rule where the optimal solution is built
up through a sequence of "local optimal" solutions. Thereby polynomiality can
only be guaranteed if this myopic decision rule is combined with an anticipant
organisation.
VII
Table of contents
Preface
PART I Preliminaries 1
Chapter 1 Terminology 2
1.1. General Notation 2
1.2. Graphs .. . . 3
1.3. Digraphs 5
Chapter 2 Linear Programming 7
2.1. Important Primal-Dual Relations 7
2.2. The Simplex Method . . . . . 9
2.3. The Primal-Dual Method ... 12
Chapter 3 Combinatorial Optimization 17
3.1. Polyhedral Combinatorics . 18
3.2. Algorithms and Complexity 23
PART II The Class of General Matching Problems 27
Chapter 4 Three Cornerstone Problems 28
4.1. Network Flow ..... . . . . 28
4.2. Graph Matching . . . . . . . . 32
4.3. Programming in Bidirected Graphs 35
Chapter 5 Important Subclasses 41
5.1. Some Special Network Flow Problems 41
5.2. Some Special Graph Matching Problems 47
5.3. Reductions for Matching Problems 52
PART III Network Flow Algorithms Revisited ........... 61
Chapter 6 Prologue: Two Apparently "Easier" Network Flow Problems 63
6.1. The Shortest Path Problem .. . . . . . . . . . . . . . . . 64
6.2. The Max-Flow Problem ........... . 70
Chapter 7 Approaches to Min-Cost Flow Problems 78
7.1. Combinatorial Analysis of Network Flow Problems
- The Primal Approach ........... . 78
7.2. Linear Programming Analysis of Network Flow Problems
- The Network Simplex Method .......... . . 83
7.3. Combinatorial Analysis of DCP - A Unifying Dual Approach . 94
7.4. Extreme Flows - The Shortest Augmenting Path Method 103
Chapter 8 Near Equivalence of Network Flow Algorithms 108
8.1. Equivalence of the Primal-Dual Method
and the Shortest Augmenting Path Method 110
8.2. Equivalence of the Network Simplex
and the Shortest Augmenting Path Method 113
PART IV Bipartite Matching Problems .......... . 119
Chapter 9 The Cardinality Matching Problem in Bipartite Graphs 121
9.1. Combinatorial Structures 122
9.2. Augmenting Path Methods 129
Chapter 10 The Assignment Problem 134
10.1. The Assignment Polytope 135
10.2. The Hungarian Method 138
10.3. A Primal Method . . . 143
10.4. The AP-Simplex Method 150
10.5. The Shortest Augmenting Path Method 158
10.6. Near Equivalence of AP-Algorithms . . 168
Chapter 11 The Hitchcock Transportation Problem 172
11.1. The Shortest Augmenting Path Approach 173
11.2. The Scaling Approach . . . . . . . . . . 178
PART V The 1-Matching Problem ..... 183
Chapter 12 The Cardinality Matching Problem 185
12.1. Combinatorial Structures . . . . . . . 185
12.2. The Blossom Algorithm ...... . 192
Chapter 13 The Min-Cost Perfect Matching Problem 200
13.1. The Matching Polytope 200
13.2. The Primal-Dual Method ..... . 208
13.3. The Primal Method 216
13.4. The Shortest Augmenting Path Approach 224
13.5. Near Equivalence of Matching Algorithms 242
13.6. Epilogue: The Matching Simplex Method 248
x
PART VI The b-Matching Problem .... 255
Chapter 14 Basic Structures and Operations 257
14.1. Combinatorial Analysis ..... . 258
14.2. The b-Matching Polytope .... . 263
14.3. Alternating Trees and Augmentations 272
14.4. Extreme b-Matchings ...... . 281
Chapter 15 b-Matching Algorithms . . . . 287
15.1. The Shortest Augmenting Path Approach 287
15.2. The Scaling Approach 293
References . . . . . . . . . 299
XI
