Table Of ContentALGEBRAIC AND COMBINATORIAL METHODS IN
OPERATIONS RESEARCH
annals of
discrete
hematics
mat
Generul Ediror
Peter L. HAMMER, Rutgers University. New Brunswick, NJ, U.S.A.
Advisory Editors
C.B ERGE, UniversitC de Paris
M. A. HARRISON, UniversityofCalifornia, Berkeley, CA, U.S.A.
V. KLEE, University of Washington, Seattle. WA, U.S.A.
J. H.VAN LINT, California Instituteof Technology, Pasadena. CA, U.S.A.
G.-t. ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A
NORTH-HOLLAND- AMSTERDAM NEW YORK OXFORD
NORTH-HOLLAND
MATHEMATlCS STUDIES 95
Annals of Discrete Mathematics( 19)
General Editor: Peter L. Hammer
Rutprs' University, New Bmnswick, MJ. U.SA
ALGEBRAIC AND
IN
COMBINATORIAL METHODS
OPERATIONS RESEARCH
Proceedings of the Workshop on
Algebraic Structures in Operations Research
Edited b y:
R. E. BURKARD
Technical University of Graz
Graz
Austria
R. A. CUNINGHAME-GREEN
University of Birmingham
Birmingham
United Kingdom
and
U. ZIMMERMANN
University of Cologne
Cologne
Federal Republic of Germany
1984
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD
Elsevier Science Publishers B.V. 1984
I
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Library of Congress Cataloging in Publication Data
Workshop on Algebraic Structures in Operations Research.
Algebraic and combinstorial methods in operations
research.
(Annals of discrete mathemstics ; 19) (North-Holland
mathematics studies ; 95)
Bibliography: p.
1. Operations research. 2. Algebras. Linear.
3. Combinatorial analysis. I. BurJmrd, Rainer E.
11. Cuninghame-Green, Raymond A., 1933.-
111. Zimmermsnn, U. (We), 1947- N. Title.
V. Series. VI. Series: North-Holland mathematics
studies ; 95.
T57.6.W67 1984 001.4'24'015125 84-13476
Ism 0-444-87571-9
PRINTED IN THE NETHERLANDS
V
FOREWORD
A recurring theme in operations research (O.R.) is that of optimization, and over the
last 35 years the subjects of O.R. and mathematical programming have developed
side by side and enriched one another.
Much traditional 0.R has been concerned with the behaviour of continuous real
variables representing e.g. material stocks, time or money, and the corresponding
Optimization theory is one in which real linear algebra, inequalities and the differen-
tial calculus have played important roles. However, many systems with which O.R. is
concerned incorporate discrete structures for which the optimization questions are
combinatorial rather than continuous: one thinks of sequencing, scheduling and
flow-problems and of the great variety of questions which can be reformulated as
path-finding circuit-finding or sub-graph-finding problems on an abstract graph.
Correspondingly, we have witnessed a vigorous growth in the theory and practice of
combinatorial optimization.
A related, but perhaps less well-known, development has been in the application of
ordered algebraic structures to optimization problems. This application is made rele-
vant by the fact that many optimization questions depend essentially on the presence
of two features: an algebraic language within which a system can be modelled and an
algorithm articulated; and an ordering among the elements which enables a signifi-
cance to be given to the concept of minimization or maximization.
By adopting this slightly abstract point of view, we can make useful reformulations:
certain bottleneck problems become algebraic linear programs; certain machine-
scheduling problems reduce to finding eigenvectors and eigenvalues of a matrix over
a semiring; certain path-finding problems reduce to the solution of linear equations
over an ordered structure.
Many optimization problems of the kind which have arisen in O.R. assume, under
such reformulation, the appearance of problems of linear algebra over an ordered sys-
tem of scalars. Hence we may look to the highly-developed classical theory of linear
algebra over the real field to give us hints as to how we might approach these problems
or, if appropriate adaptations of classical techniques cannot be found, we have a
well-defined research program to elucidate the theory of linear algebra over such
ordered structures, and to see how far the algorithms and duality principles, familiar
to us from linear and combinatorial optimization over the real field, extend to more
general structures.
vi Foreword
These questions have stimulated a good deal of research over the last twenty-five
years. From a few isolated publications by one or two researchers in the late 1960’s
the subject has matured into an identifiable branch of applicable mathematics, with
an international following. We invited a number of those who have contributed to
this development, to participate in the production of a publication featuring some
of their more recent work. It is the result of their enthusiastic acceptance of this invk
tation which we are now pleased to present as this volume in the series of Annals of
Discrete Mathematics.
R.E. Burkard
R.A. Cuninghame-Green
U. Zimmermann
Acknowle&ement: I should like to add a personal note of gratitude to Tricia Carr,
who made such a beautiful job of preparing the manuscript.
R.A. Cunjnghame-Green
CONTENTS
Foreword v
J. ARAOZ, Packing problems in semigroup programming 1
P. BRUCKER, A greedy algorithm for solving network flow problems
in trees 23
P. BRUCKER, W. PAPENJOHANN, and U. ZIMMERMANN, A dual
optimality criterion for algebraic linear programs 35
BUTKOVIC,
P. On properties of solution sets of extremal linear programs 41
R.A. CUNINGHAME-GREEN, Using fields for semiring computations 55
R.A. CUNINGHAME-GREEN and W.F. BORAWITZ, Scheduling by
non-commutative algebra 75
Ck EBENEGGER, P.L. HAMMER, and D. DE WERRA, Pseudo-boolean
functions and stability of graphs 83
R. EULER, Independence systems and perfect k-matroid-intersections 99
U. FAIGLE, Matroids on ordered sets and the greedy algorithm 115
A. FRANK and E. TARDOS, An algorithm for the unbounded matroid
intersection polyhedron 129
A.M. FRIEZE, Algebraic Flows 135
M. GONDRAN, and M. MINOUX, Linear algebra in dioids: a survey of
recent results 147
H.W. HAMACHER and S. TUFEKCI, Algebraic flows and time-cost
tradeoff problems 165
H.W. HAMACHER, J.-C. PICARD, and M. QUEYRANNE, Ranking the
cuts and cut-sets of a network 183
P. HANSEN, Shortest paths in signed graphs 201
B.L HULME, A.W. SHIVER,a nd P.J. SLATER, A boolean algebraic
analysis of fae protection 215
B. MAHR, Iteration and summability in semirings 229
RH. MOHRING, and F.J. RADERMACHER, Substitution decomposition
for discrete structures and connectionsw ith combinatorial
optimization 25 7
K. ZIMMERMA", On max-separable optimization problems 357
U. ZIMMERMA", Minimization of combined objective functions on
integral submolar flows 363
ANNALS OF DISCRETE MATHEMATICS
Vol. I: Studies in Integer Programming
edited by P. L. HAMMER, E. L. JOHNSON, B. H. KORTE and
G. L. NEMHAUSER
+
1977 viii 562 pages
Vol. 2: Algorithmic Aspects of Combinatorics
edited by B. ALSPACH, P. HELL and D. J. MILLER
1978 out of print
Vol. 3: Advances in Graph Theory
edited by B. BOLLOBAS
+
1978 viii 296 pages
Vol. 4: Discrete Optimization, Part I
edited by P. L. HAMMER, E.L. JOHNSON and B. KORTE
+
1979 xii 300 pages
Vol. : Discrete Optimization, Part I1
edited by P. L. HAMMER, E.L. JOHNSON and B. KORTE
+
1979 vi 454 pages
Vol. 6: Combinatorial Mathematics, Optimal Designs and their
Applications
edited by J. SRIVASTAVA
+
1980 viii 392 pages
VOl. 7: Topics on Steiner Systems
edited by C. C. LINDNER andA. ROSA
+
I980 x 3jO pages
VOl. 8 : Combinatorics 79, Part I
edited by M. DEZA and I. G. ROSENBERG
+
I 980 xxii 3 I0 pages
Vol. 9: Combinatorics 79. Part I1
crlifcdb y M. DEZA rind 1. G. ROSENBERG
+
I980 viii 3 I 0p ages
