PTIMA
MATHEMATICAL PROGRAMMING SOCIETY NEWSLETTER
Number 17
December 1985
Society Prizes Awarded At Boston Meeting
Dick Cottle Resigns, Mike
Todd New EditorInChief
Publication Matters at the Boston
Symposium
After a term of six years as a
distinguished EditorinChief of Mathe
matical Programming and the Mathe
matical Programming Studies, Richard W.
Cottle has decided to resign. The Council
of the Society has expressed its sincere
gratitude to Dick for his outstanding
contributions to the Society in this central
function.
It is a pleasure to announce the
appointment of Michael J. Todd as our
new EditorinChief. Mike will assume
editorial responsibility for the Journal and
the Studies on January 1, 1986, for a term
of three years, which is renewable. In
cooperation with the Publications Com
mittee of the Society, Mike will make
recommendations on various crucial issues
related to the Journal and the Studies.
One of these issues is the publication
frequency of the Journal. From the
beginning in 1971, we published two
volumes (six issues) per year. In 1982
this was increased to three volumes per
year as a remedy for unacceptable
backlogs. It helped; the backlog is now at
an alltime minimum of six months.
While this is attractive for our authors, it
can be unnerving from an editorial point
of view. A return to the original publica
tion frequency might eventually be
unavoidable. Please view this as an
invitation to submit!
Another issue is the status of the
Studies and their relationship to the
Journal. A Study is "devoted to a unified
continued on page 11
William OrchardHays with Michael Saunders
First OrchardHays Prize
to Michael Saunders
The first OrchardHays Prize was
awarded at the Boston meeting to
Michael Saunders for his work in
computational mathematical program
ming. Saunders is at the Stanford Opti
mization Laboratory, Department of
Operations Research, Stanford University.
The award is administered by the Society's
Committee on Algorithms.
The award citation reads: "On this
fifth day of August 1985, for his
innovative mathematical work which has
advanced significantly the design and
implementation of nonlinear program
ming algorithms, and for establishing the
standard in versatile, reliable and available
software in this field, initially through
early versions of MINOS done jointly
with B. Murtagh, and more recently
through a variety of projects with P. Gill,
W. Murray and M. Wright."
Karla Hoffman
Ellis Johnson and
Manfred Padberg Awarded
1985 Dantzig Prize
The 1985 Dantzig Prize was awarded
to Ellis L. Johnson and Manfred W.
Padberg at the opening ceremonies of
the Boston MPS Symposium.
The Dantzig Prize "is awarded jointly
by the Mathematical Programming So
ciety and the Society for Industrial and
Applied Mathematics. The prize is
awarded for original work which, by its
breadth and scope, constitutes an out
standing contribution to the field of
mathematical programming.... The contri
bution(s) for which the award is made
must be publicly available and may
belong to any aspect of mathematical pro
gramming in its broadest sense. The
contributions eligible for consideration are
not restricted with respect to the age or
number of their authors although prefer
ence should be given to the singly
authored work of 'younger' people".
The first Dantzig Prize was awarded in
1982 jointly to M.J.D. Powell and R.T.
continued on page 2
Beck, Lenstra and Luks
Receive Fulkerson Awards
The three recipients of the Fulkerson
Prize awarded at the International
Symposium at MIT in August were as
follows:
* to Jozsef Beck, Mathematical
Institute of the Hungarian Academy of
Sciences, H1395 Budapest, Pf. 428,
Hungary, for the paper "Roth's Estimate
of the Discrepancy of Integer Sequences is
Nearly Sharp," Combinatorica 1 (4)
319325 (1981).
continued on page 3
~~"8" s"s~s~"s"s""a~"l%~"~~""~^~la"
IBPEls~
II ~I '~ "I~
1985 Dantzig Prize ___
J____1985 Dantzig Prize
L to R: Manfred Padberg and Ellis Johnson
with George Dantzig
Rockefellar. The 1985 Committee (G.
Nemhauser, M. Balinski, M. Powell and
R. Wets, chairman) formulated its recom
mendation in the following terms.
A rich variety of problems, including
those of resource allocation, distribution,
facility location, scheduling, production
and reliability, can be formulated as
discrete optimization models. Generally,
these models involve a large number of
variables and/or constraints. Although the
models are very robust in the situations
that they can describe, their efficient
solution has proven to be a very difficult
problem in computational mathematics.
Only recently have the theory and compu
tational aspects of the subject been
synthesized to achieve algorithms that are
capable of solving discrete optimization
models of significant size and complexity.
Johnson and Padberg, alone and together,
have made major theoretical and computa
tional contributions to the recent signi
ficant developments in discrete optimiza
tion and have applied their work to solve
practical problems.
Modern integer programming began
thirty years ago. In a seminal paper,
Dantzig, Fulkerson and Johnson solved a
48city traveling salesman problem by
linear programming. Their essential idea
was to use valid linear inequalities or
cutting planes to cut off optimal fractional
solutions to a linear programming relaxa
tion of the traveling salesman problem.
Their approach, however, was ad hoc.
The method for generating valid inequal
ities was neither general nor automatic.
The contribution was an idea rather than a
general procedure and a demonstration that
the idea could work.
In 1958, Gomory gave a finite cutting
plane algorithm for the solution of integer
programming problems. Here the cutting
planes are automatically generated and it
was shown constructively that the general
integer programming problem can be
solved as a finite sequence of linear
programs. Unfortunately, Gomory's basic
algorithm and some variations failed in
practical computation because of ex
tremely slow convergence. The reason for
this is that the cutting planes used by the
algorithm are weak. As a result,
branchandbound algorithms became the
main practical approach to solving, often
approximately, integer programming
problems. But, because of its enumer
ative character, the success with this
approach has been unpredictable and the
size of problems that can be solved is
severely limited.
In 1965, Edmonds showed that a
special integer programming problem,
known as the matching problem, could be
solved very efficiently using a linear
inequality description of the convex hull
of integral solutions. This approach has
proven to be enormously valuable and
seminal in the development of a literature
known as polyhedral combinatorics. The
main obstacle that prevents this approach
from being applied to general integer
programs, and most practically important
special cases, is that an explicit linear
inequality description of the convex hull
of integral solutions is unknown and is
unlikely to be found. Nevertheless, a
significant effort has been made to identify
and use facets (full dimensional faces) of
the convex hull of feasible integer
solutions to many problems, including set
covering and packing problems, the
knapsack problem, the traveling salesman
problem, fixedcharge network problems
and relaxations of the general integer
programming problem. The basic idea is
to design cutting plane algorithms that
use strong cuts, frequently facets. The
challenge is to characterize them and show
how they can be automatically generated
to render fractional points infeasible.
In this endeavor, Johnson and Padberg
played central roles, Johnson primarily in
his work on the group theoretic approach
to the general integer programming prob
lem (work begun by Gomory in the late
1960's), and Padberg primarily in his
work on structured packing and covering
problems and the traveling salesman
problem.
Johnson, initially with Gomory, in a
series of papers beginning in 1972,
pursued the idea of generating facets of the
convex hull of feasible solutions to the
grouptheoretic relaxation of an integer
program, i.e., the socalled comer poly
hedra. This is one of the most general
family of polyhedra for which the convex
hull of integral solutions has been
characterized. He pioneered the use of
subadditive functions in the generation of
strong cutting planes. Several of his
papers have significant computational
experiments that demonstrate the effi
ciency of the algorithms. Johnson has
also worked on a variety of combinatorial
optimization problems including edge
coloring, matching, chinese postman,
scheduling and plant location. In 1966,
he wrote a fundamental paper on labeling
techniques for network flow problems that
led to the data structures used in modem
network flow codes. And while most of
his research is in discrete optimization, he
has also worked in inventory theory.
Padberg, on the other hand, began by
focusing on special problems, especially
packing and covering problems, and the
traveling salesman problem. His main
thrust has been the study of facets and
their use in developing efficient algo
rithms. Beginning in 1972, he published
a number of papers on the facial structure
of set packing polyhedra. Alone, and in
joint work with Grotschel and Rao, he has
produced many results on the traveling
salesman polytope. This work has been
brought to fruition in two computational
studies, one with Crowder and the other
with Hong, in which they systematized
the early work of Dantzig et al. on the
traveling salesman problem by showing
how to automate the generation of strong
cutting planes. Their strong cutting
plane/branchandbound algorithm has
solved traveling salesman problems with
up to 318 cities. Padberg, in joint work
with Wolsey and others, has applied this
approach successfully to fixedcharge flow
~~ I
problems. Padberg has also made signi
ficant contributions to the characteriza
tion of perfect graphs. In joint work with
Rao, he has shown how the ellipsoid
method can be used to obtain polynomial
time algorithms for combinatorial opti
mization problems. (These important
results were obtained independently by
Grotschel et al.)
The major streams of Johnson's and
Padberg's theoretical and computational
work on strong cutting planes merged in
their joint 1983 paper with Crowder. In
it, the facets developed earlier for knapsack
polytopes were linked with linear pro
gramming and branchandbound to pro
duce the code for 01 integer programs
known as PIPX. This code has solved
largescale, realworld, 01 problems with
up to 2750 variables and represents state
oftheart methodology for 01 integer
programs.
The citation reads: "The 1985
George B. Dantzig Prize is awarded jointly
to Ellis L. Johnson and Manfred W.
Padberg. Both have made very significant
contributions to the theory, computation
and uses of discrete optimization. Begin
ning at different sources, Johnson with the
group theoretic relaxation of integer pro
gramming, and Padberg with covering and
packing problems and the traveling sales
man problem, have initiated the strong
cutting plane approach to solving integer
programming problems. This work has
produced a stateoftheart algorithm for
01 integer programs and has also contrib
uted substantially to the theory of integral
polyhedra, i.e., the description (or partial
description) of the convex hull of integral
points by a set of linear inequalities.
Each individually has made numerous
contributions to graph theory, combina
torial optimization and applications of
mathematical programming. Their overall
research efforts have combined mathe
matical insight, computational develop
ment and application very much in the
spirit of George Dantzig's approach and in
the highest standards of mathematical
programming."
M. Balinski
R. Wets
Fulkerson Awards L
Citation: In the cited paper Beck
solves a longstanding problem of K.F.
Roth by showing that the integers from 1
to N can be 2colored so that in each arith
metic progression the numbers of red and
blue points differ by less than N1/4 +e.
He proves this by deriving a general upper
bound on the discrepancy of a family of
sets, which is defined as the minimum,
over all twocolorings of the elements, of
the largest imbalance between the two
colors in a set from the given family.
In further papers Beck solves problems
raised by K.F. Roth, W.M. Schmidt and
others. He determines with great precision
the discrepancies of families defined by
disks, rectangles and convex planar sets.
His methods involve an ingenious com
bination of combinatorial arguments with
methods from linear algebra, probability
theory and Fourier analysis. The recent
growing interest in combinatorial discrep
ancy theory has been inspired in large part
by his work.
Fulkerson Awardee H.W. Lenstra Jr..
Jozsef Beck and Eugene Beck also received
Fulkerson Awards (photos not available).
* to H.W. Lenstra, Jr., Department
of Mathematics, University of Amster
dam, Mathematisch Instituut, Roeters
straat 15, 1018 WB, Amsterdam, The
Netherlands, for the paper "Integer Pro
gramming with a Fixed Number of Vari
ables, "Mathematics of Operations Re
search 8 (4), 538548 (1983).
Citation: The main result of the cited
paper is that the problem of integer linear
programming can be solved in polynomial
time if the number of variables is fixed.
This theorem answers a longstanding
open question. Of particular interest are
the methods introduced by Lenstra in order
to solve this problem. His algorithm is
a combination of two important methods.
First, he develops a technique which en
ables him to extract, as much informa
tion from the set of real solutions of the
program as necessary. This step is quite
interesting from the point of view of con
vex analysis and is in a sense polar to the
ellipsoid method. In the second phase of
the algorithm, he takes integrality into
account, using a "basis reduction" tech
nique based on the geometry of numbers.
This establishes a connection between
integer programming and classical fields
of number theory.
The result has already found many
applications. It has been used to show
that best simultaneous diophantine ap
proximation for a fixed number of reals
can be found in polynomial time. It was
also used to break a version of the Merkle
Hellman cryptosystem. It has inspired a
great deal of further research in the algo
rithmic theory of lattices of points, in
diophantine approximation ahd in integer
linear programming.
to Eugene M. Luks, Department of
Computer and Information Sciences,
University of Oregon, Eugene, Oregon
97403, for the paper "Isomorphism of
Graphs of Bounded Valence Can Be Tested
in Polynomial Time," Journal of Com
puter and System Sciences 25 (1) 4265
(1982).
Citation: The problem of determining
whether two graphs are isomorphic is one
of the most fundamental problems in
computational graph theory, and its com
putational complexity has attracted a great
deal of interest. In the cited paper Luks
provides a polynomialtime algorithm for
the previously intractable case of bounded
degree graphs. Luks' method is to reduce
this problem to the set stabilizer problem
for certain permutation groups and then to
introduce unexpectedly efficient divideand
conquer methods to solve the problem for
this class of groups. The links that Luks
forged enabled deep methods from group
theory to be applied to the graph isomor
phism problem and inspired significant
new research by Luks and others on the
computational complexity of group
theoretic problems.
~ ~ ~
CONFERE
NCE NOTE
International Conference on
Stochastic Programming
Prague, Czechoslovakia
September 1519, 1986
This conference is being organized by
the Faculty of Mathematics and Physics,
Charles University Prague, cosponsored
by the International Institute for Applied
Systems Analysis, Laxenburg, and the
Committee for Stochastic Programming of
the Mathematical Programming Society.
The preregistration should be mailed to
the following address by December 15,
1985:
Dr. Tomas Cipra
Dept. of Statistics
Sokolowska 83
18600 Prague 8
Czechoslovakia
Jitka Dupacova
Joint US/FRG Seminar
Applications of
Combinatorial Methods in
Mathematical Programming
Gainesville, Florida
March 1822, 1985
Many areas of Mathematical Pro
gramming use methods developed in the
field of combinatorics. The purpose of
this conference, organized by A. Bachem
(Koeln) and H. Hamacher (Gainesville),
was to bring together leading experts of
the United States and the Federal Republic
of Germany to discuss modern trends and
new results in combinatorics. There were
also participating researchers from six
nonUS/FRG countries.
The topics discussed included the
analysis of NPcomplete problems and
data structures for polynomially solvable
problems, investigations of combinatorial
structures and the modelling of real world
problems. A large portion of the con
ference was devoted to discussions of open
problems which were provided by every
participant. Abstracts and open problems
are published as reports of the Department
of Industrial and Systems Engineering,
University of Florida, and the Department
of Mathematics, University of Koeln, and
are available upon request.
The social program included a
reception, with a welcoming speech by
the President of the University of Florida,
and a hike.
H. Hamacher
14th International Symposium
on Mathematical
Programming 1991
The Mathematical Programming Soci
ety is seeking a host for its International
Symposium in 1991. For information
please contact the chairman of the Sym
posium Advisory Committee by mail or
phone as follows:
M. Grotschel
Institute f. Mathematik
Univ. Augsburg
Memminger Str. 6
D8900 Augsburg
W. Germany
Phone: (0821) 598317
Callfor Papers
Optimization Days 1986
April 30 May 2, 1986
Montreal, Canada
OPTIMIZATION DAYS 1986 will be
held at Ecole des Hautes Etudes Com
merciales de Montreal (Graduate Business
School of the University of Montreal)
from April 30 to May 2, 1986.
The topics include all aspects of opti
mization theory and applications. Con
tributed papers should be sent before
January 31, 1986 to:
Alain Haurie
Director of GERAD
Ecole des H.E.C.
5255 avenue Decelles
Montreal (Quebec) H3T 1V6
Tel: (514) 3406042
New Doctoral Program
Offered
European Doctoral Program in
Quantitative Methods in Manage
ment A Joint Program
The Universite ParisDauphine,
Erasmus Universiteit Rotterdam
and Universite Catholique de
Louvain have set up a coordinated
doctoral program in quantitative
methods in management. Each
student will be registered in a
doctoral program of one of the
universities and will typically spend
the second year of his doctoral
studies at one of the other two.
The program is aimed at a
limited number of highly qualified
students with a fundamental interest
in management and a good back
ground in quantitative methods.
Further information about the
program can be obtained from:
Professor J.M. Lasry, UER
Mathematiques de la Decision, Uni
versite Paris IX, Place Marechal de
Lattre de Tassigny, F75775 Paris,
Cedex 16, France.
Professor A.H.G. Rinnooy
Kan, Econometric Institute, Eras
mus University, P.O. Box 1738,
3000 DR Rotterdam, The
Netherlands.
Professor L.A. Wolsey,
Industrial Engineering, CORE,
Professor Ch. Delporte, School of
Management, IAG, Universite
Catholique de Louvain, B1348
LouvainLaNeuve, Belgium.
L. Wolsey
_ I_ ~ _1_1_ _1_ ~
Applications of Nonlinear Programming to.
Optimization and Control
Edited by H.E. Rauch
Pergamon Press, Oxford
1984
iSHN 0080305741
This is a very nice collection of interesting papers on the
application of Nonlinear Programming Methods to Practical
Control Problems.
These papers were presented at the 4th IFAC Workshop in San
Francisco, which was organized by the editor on June 2021,
1983. The members of the corresponding Program Committee
were Professors Arthur Bryson, Jr. (U.S.A.), H.T. Banks
(U.S.A.), Phillip Gil (U.S.A.), Faina M. Kirillova
(U.S.S.R.), R.W.H. Sargent (Great Britain) and J.P. Yvon
(France).
This volume contains contributions of 17 international
experts who presented their latest work in this field. The papers
cover a wide range of research topics starting with Computer
Aided Design of Practical Control Systems, continuing through
advanced work on quasiNewton methods and gradient restoration
algorithms and culminating with specific examples which apply
these methods to representative problems. Many examples are
presented.
Study of these papers is highly recommended. This book is
an essential contribution for the application of nonlinear
programming methods to modem practical control problems.
D. Pallaschke
Algebraic and Combinatorial Methods in
Operations Research
Edited by R.E. Burkard, R.A. CunninghameGreen,
and U.Z. Zimmermann
NorthHolland, Amsterdam
1984
ISBN 0444875719
In recent years various methods have been applied to describe
and solve discrete optimization problems. For example, by
choosing the cost coefficients as elements of an ordered algebraic
structure, a unification and generalization of various problems
(including sum, bottleneck and lexicographical objectives) can be
achieved. Another example provides the maxmin algebra which
leads to applications in scheduling and fuzzy equations. A
workshop in Bad Honnef, April 1982, brought together for the
first time researchers from different countries to discuss the
applications of algebraic methods to Operations Research
problems. The volume contains 20 selected contributions to this
workshop.
The papers contain original work and can be grouped as
follows:
Several contributions consider algebraic flow problems
proving general max flowmin cut theorems. Also algorithms to
solve the algebraic flow problems are given.
Another group of papers deals with new results on matroids,
matroid intersections and the Greedy Algorithm. In particular,
new results on perfect independence systems and matroids on
ordered ground sets are presented.
Several papers investigate algebraic linear programs. In one a
dual optimality criterion is generalized; another paper exploits
solution properties of extremal linear programs.
Two papers investigate graph problems. The shortest path
problem on signed graphs is considered. The other paper
investigates relations between pseudo Boolean functions and the
stability number of graphs.
Also included is a thorough investigation of the relationship
between substitution decomposition known for Boolean functions,
set systems and relations as applied to optimization problems. A
survey paper provides recent results of the french school on
computations in dioids.
Finally, new applications of the algebraic approach are also
presented. One paper deals with a fire protection problem;
another considers scheduling problems using noncommutative
algebra.
To summarize, the contributions in the book give insight into
the state of the art of the algebraic side of combinatorial optimi
zation. The book is recommended to anyone interested in recent
trends in discrete optimization.
F. Rendl
Optimization Theory and Applications
by Jochen Werner
Vieweg, Wiebaden
1984
ISBN 3528085940
The book gives a nice introduction to the foundations of
optimization theory and its applications. It is the refined and
enlarged version of the manuscript of a course the author gave
at the University of Gottingen. This origin of the book
strongly influences the lively style of the presentation. A
number of accompanying examples, to which the author refers
throughout, serve as a permanent stimulation. The book can be
recommended either for selfstudy or as a textbook for an
introductory course in optimization.
The book is selfcontained and does not require more than
elementary knowledge from a first course on algebra and
analysis. All tools from functional analysis needed in the text
(like separation theorems) are developed in the book. The
most important aspects of the course are the duality theory for
convex programming and necessary optimality conditions for
nonlinear optimization problems in infinitedimensional
spaces. No numerical methods are discussed; however, it is
made clear that many of the results presented are fundamental
for algorithms that are applied in practice to the numerical
solution of optimization problems.
After some examples in Chapter 1, the classical duality
theory for finitedimensional linear programming problems is
developed (Chapter 2); the approach stresses the geometrical
background. Chapter 3 provides some tools from functional
analysis which are used in Chapters 4 and 5 to discuss the
main results in convex and in differentiable optimization in
normed linear spaces. The classical optimality conditions are
derived and applied to a number of control and approximation
problems.
Jochem Zowe
__~
~ ~
Book Reviews
Sensitivity, Stability and Parametric Analysis
Mathematical Programming Study 21
Edited by A.V. Fiacco
North Holland, Amsterdam
1984
Several years ago the editor, Prof. A.V. Fiacco, had a
splendid idea, to organize yearly a special conference on Mathe
matical Programming with Data Perturbations. These conferences
have been highly acknowledged by specialists the world over and
the corresponding proceedings have sold well.
Fiacco organized and edited this Mathematical Programming
Study which is devoted to stability analysis and parametric
programming as well as to mathematical programming with data
perturbations. It shows convincingly that sensitivity analysis,
parametric programming, mathematical programming with data
perturbations, for which there is no specialized Journal or no SIG,
comprise a branch of operations research which can no longer be
overlooked.
This book can be recommended to all scientists working in
the area of the theory of linear and nonlinear pr,,er.lin'i;r g, in
perturbation analysis, to specialists working in the area of
inventing new methods for solving nonlinear programs, and to
students of OR.
T. Gal
Data Structures and Algorithms
Volumes I, II and III
By Kurt Mehlhorn
Springer Verlag, Berlin
1984
During the last decade considerable progress has been made in
the area of data structures and efficient algorithms. Many of the
recent inventions, e.g. randomized algorithms, new methods for
proving lower bounds or new data structures for weighted dynamic
data, have never appeared in book form before.
The author's work is split into three volumes. Volume I
starts with a thorough treatment of different basic models of
computation and introduces most of the commonly used data
structures. The very precise presentation is accompanied by an
easily understood intuitive description in order to enable the
nonexpert to capture the basic ideas. Following the introductory
section, various sorting algorithms are treated extensively and
lower bounds for the sorting problem are derived. Together with
somewhat more sophisticated data structures, the results are used
to obtain efficient methods for onedimensional searching. The
great effort spent on discussing sorting problems is justified by
its practical importance: As the author remarks, an IBM estimate
claims that sorting consumes about 25% of the total computing
time spent by commercial codes.
Volume II is about Graph Algorithms. The author discusses
how graphs can be represented in a computer and treats dillcrnti
methods for exploiting a graph (such as depth first and breadth
first search). He intends to explain the principles of how to deal
with graphs from a computational point of view and therefore
restricts himself to considering only those problems which are
quite "simple" from a theoretical point of view. For example, the
matching problem is treated for bipartite graphs only. A
considerable part is devoted to developing the relationship
between path problems in graphs and matrix multiplication. The
last chapter of this book is an excellent introduction to the
theory of NPcompleteness.
Volume III starts with considering multidimensional search
problems. ddimensional search trees are shown to be, in a sense,
the optimal data structures for this kind of problem. General
principles are described which might be used to overcome in part
the problem of balancing :1milh Iliil;i'lr.Il search trees. The
major part of this volume is devoted to computational geometry,
which deals with the problem of determining the intersection of
geometric objects such as lines or polygons in the plane. Many
logarithmic algorithms are developed to solve these problems.
The material covered in this book leads the reader from the
basics to current research. The presentation is very clear and
everything is developed in a straightforward manner. For those
who are familiar with the fundamentals of data structures, such as
queues, stacks and linked lists, volumes II and III are recommended
separately.
W. Kern
Algebraic and Geometric Combinatorics
Annals of Discrete Mathematics Vol. 15
Edited by Eric Mendelsohn
North Holland, Amsterdam
1982
ISBN 0444863656
The theories of designs, graphs, latin squares, finite
groups, geometries and lattices, which represent the main subjects
treated in this volume, constitute cornerstones of combinatorial
mathematics. The papers on design theory deal with the
construction of block designs, the study of Steiner systems and
their generalizations, or the enumeration of special types of
designs. The hamiltonicity of metacirculant graphs, studies of
rigid as well as distance regular graphs, and two new proofs of the
MendelsohnDulmage theorem, one within the framework of linear
inequalities, are the contents of the graphtheoretical papers. Two
classical problems related to latin squares are the establishment of
necessary and sufficient conditions for the completeability of
incomplete latin squares and the existence of special types of
latin squares. Of particular interest are implications for
timetabling (here "matchtabling") and, more theoretically, the
treatment within universal algebra or the field of finite groups and
geometries. Within this latter area a study of the product of all
elements in a finite group, a discussion of finite fields from a
combinatorial point of view, incidencegeometric aspects of finite
abelian groups and the construction of special partial geometries
are presented. The concept of lattice polyhedra finally has proven
to be very suitable in embracing various results from polyhedral
combinatorics. Properties, constructions and examples for such
polyhedra are the subject of a first contribution within lattice
theory. A second paper displays the use and appeal of lattice
diagrams.
This book is more than just a collection of papers. It is the
presentation of interrelations between several branches of
combinatorics which makes it interesting to read. It also reflects
the strong influence on the development of algebraic and
geometric combinatorics of Prof. Nathan Mendelsohn, to whom
this book has been dedicated on the occasion of his 65th
birthday.
Reinhardt Euler
 ~~ ~
ook Reviews
Book Reviews
Combinatorial Algorithms
by T.C. Hu
AddisonWesley, New York
1982
ISBN 0201038595
The author opens by stating his intent to present "some
combinatorial algorithms" common to computer science and
operations research. His final remark is, "Hopefully, this book
will arouse interest in combinatorial algorithms." He succeeds in
providing a readable, interesting account of some combinatorial
algorithms.
The informal development and motivation can be enjoyable as
well as useful as a pedagogical aid. The presentation style uses
illustrative examples instead of general descriptions. On the
topics covered, bibliographical notes and references are provided.
The author dispenses with the usual onslaught of definitions
of terms and notation. Consequently, he relies in part on the
reader's intuition and tolerance of imprecision, which could be
frustrating to the beginner. (For example, a path is defined as a
directed path and then is later allowed to have reverse arcs.)
The book begins with a complete treatment of the shortest
path problem. The next topic covered is the maximum flow
problem, where the maxflowmincut theorem is presented and
several methods of computing a maximal flow as part of the
FordFulkerson algorithm are described. A scant treatment of the
minimum cost flow problem is provided. Considerable attention
is paid to the problems of realization, analysis, and synthesis of
multiterminal maximum flow problems, PERT, and the optimum
communication spanning tree problem. Dynamic programming
is discussed, illustrated by the shortest path problem, the
knapsack problem, and the minimum cost alphabetic tree
problem. Backtracking and branch and bound are presented and
illustrated, followed by algorithms for construction of binary trees
of minimum weighted path length and optimal alphabetic trees.
Heuristic algorithms are given for the coin changing problem, bin
packing, job scheduling, and partitioning a convex polygon into
triangles. The book concludes with a chapter on complexity
theory, including a brief guide to facing new combinatorial
problems.
With an informal treatment of selected topics, this book gives
the reader a taste of the field of combinatorial algorithms.
C.R. Coullard
Linear Programming
by V. Chvatal
Freemann, New York
1983
ISBN 0716711958
It has been my opinion that, in spite of a plethora of books
on the subject, the best of the textbooks in linear programming
written in the early sixties was not significantly improved upon
in the succeeding 20 years. Obviously, a quantum leap in quality
was inevitable. It has come with this brilliant new exposition of
Chvatal.
This book is exceptionally well written. Few authors can
write this well, and of those who can, fewer take the trouble. The
author is to be commended for the obvious pains he has taken in
attention to detail and style. It is most apparent in the
background chapter on linear algebra, in numerous references
explaining historical aspects, and in careful and complete
presentations of advanced topics like modem basis factorizations
and the ellipsoid method.
An interesting aspect of the book is that, while ultimately
quite sophisticated, it is completely selfcontained. The book
assumes no linear algebra, presenting most of the basic ideas and
theory of the simplex method and duality from scratch in the first
five chapters. Proofs are algebraic and constructive in spirit.
There is no use, even for motivation, of either geometry or
convexity.
Chapter Six, Gaussian Elimination and Matrices, is a tour de
force of linear algebra, in preparation for the introduction of the
revised simplex method. A reader who does come to this book
with no linear algebra background will have to take this chapter
slowly, but it is complete, covering precisely the right topics
with the right emphases. On the other hand, more sophisticated
readers will still learn a lot from this chapter. I did, and I suspect
that authors of the current crop of linear algebra texts would also.
Chvatal's treatment of the revised simplex method is based on
keeping the basis as a product of eta matrices, plus periodic
refactorization into triangular factors. The author does an
excellent job of presenting the methods and explaining why the
typical sparse structure of problems solved in practice dictates
these strategies. The remainder of this first section extends
previous results to more general forms of linear programs and
treats sensitivity analysis, including the dual simplex method.
The second section, Selected Applications, ranges from
concrete elementary applications like allocation of resources
through the cutting stock problem and matrix games 'o theoretical
chapters on systems of inequalities and geometry. In the latter
chapters linear programming is used in an elegant way to derive
deep and beautiful mathematical results.
The third section is on network flow problems. Here another
huge improvement on standard presentations is evident. The
network simplex method is described in a way that is at once
more general, more direct, and closer to modern computer
implementations than those in the standard textbooks. (Can we
hope that we have finally seen the last of transportation
tableaux?) It is incredible to me that this algorithm, arguably the
most important in all of combinatorial optimization, is not even
described in some books on that subject. This section gives a
number of nice combinatorial applications of the network simplex
method as well as describing the "labelling methods." While the
latter are well done and completely up to date, to me their
inclusion seems unnecessary.
The last section is on advanced techniques, including more
sophisticated methods for the revised simplex method, and
describes generalized upper bounding and DantzigWolfe
decomposition. Again, the presentations are not standard and are
very well done. An appendix describes the ellipsoid method
explaining the theory as well as placing it in an historical and
practical perspective.
This book is not easy to criticize. I was a little disappointed
and surprised that the author did not find occasion to introduce
some integer programming and cutting planes somewhere,
especially given his own research interests. As for errors, they
seem hard to find; I can report three. On page 113 "revised
simplex method" should read "standard simplex method." On page
400 the attribution to Munkres of an O(n3) assignment algorithm
is not accurate. Finally, not all of the exercise solutions on page
466 are correct.
W.H. Cunningham
_~I~
Technical Reports & Working Papers
Stichting Mathematisch Centrum
Centrum voor Wiskunde en Informatica
Postbus 4079 1009 AB Amsterdam
Kruislaan 413 1098 SJ Amsterdam
J. K. Linstra and A.H.G. Rinnooy Kan, "New Directions in
Scheduling Theory," OSR8401.
J.P.C. Blanc, "Asymptotic Analysis of a Queueing System
with a TwoDimensional State Space," OSR8402.
J. Han and M. Yue, "A Study of Elimination Conditions for
the Permutation FlowShop Sequencing Problems," OSR8403.
J.H. van Schuppen, "Overload Control for an SPC
Telephone Exchange: An Optimal Stochastic Control
Approach," OSR8404.
M.W.P. Savelsbergh and P. van Emde Boas, "BOUNDED
TILING, an Alternative to SATISFIABILITY?" OSR8405.
P.J.C. Spreij, "An OnLine Parameter Estimation Algorithm
for Counting Process Observations," OSR8406.
J.P.C. Blanc and E.A. van Doom, "Relaxation Times for
Queueing Systems," OSR8407.
J.P.C. Blanc, "The Transit Behaviour of Networks with
Infinite Server Nodes," OSR8408.
M.W.P. Savelsbergh, "Local Search in Routing Problems
with Time Windows," OSR8409.
E.A. van Doom, "A Note on Equivalent Random Theory,"
OSR8410.
R.M. Karp, J.K. Lenstra, C.J.H. McDiarmid, "Probabilistic
Analysis of Combinatorial Algorithms: An Annotated
Bibliography," OSR8411.
The Ohio State University
Department of Industrial and Systems Engineering
1971 Neil Avenue
Columbus, OH 43210
C.H. Reilly and C.C. Petersen, "PIPZI: A Partial Enumer
ation Algorithm for Pure 01 Polynomial Integer Programming
Problems," 1984013.
C.H. Reilly and C.C. Petersen, "Imbedded Linear Programs in
Nonlinear Integer Programming Problems," 1984014.
The Johns Hopkins University
Department of Electrical Engineering
and Computer Science
Baltimore, MD 21218
G.G.L. Meyer, "An AntiJamming Procedure for Relaxation
Algorithms," 84/01.
J. O'Rourke, "Finding Minimal Enclosing Tetrahedra," 84/05.
J. O'Rourke, A. Aggarwal, S. Madilla and M. Baldwin, "An
Optimal Algorithm for Finding Minimal Enclosing Traingles,"
84/08.
W.J. Rugh, "Design of Nonlinear Compensators for Non
linear Systems by an Extended Linearization Technique," 84/10.
J. O'Rourke, H. Booth and R. Washington, "Connect
theDots: A New Heuristic," 84/11.
J. O'Rourke, "Finding Minimal Enclosing Boxes," 84/14.
Baumann and W.J. Rugh, "Feedback Control of Nonlinear
Systems by Extended Linearization," 84/15.
Northwestern University
Department of Industrial Engineering
and Management Sciences
Evanston, Illinois 60201
R. Fourer, "A Simplex Algorithm for PiecewiseLinear
Programming I: Derivation and Proof," Technical Report 8502,
February 1985.
R. Fourer, "A Simplex Algorithm for PiecewiseLinear
Programming II: Finiteness, Feasibility and Degeneracy," Tech
nical Report 8503, February 1985.
A.P. Hurter, "Production Location Problems with Demand
Considerations," February 1985.
A.P. Hurter and P. Lederer, "Competition of Firms:
Discriminatory Pricing and Location," August 1984.
A.P. Hurter and P. Lederer, "Spatial Duopoly with Dis
criminatory Pricing," March 1985.
A.P. Hurter and J. Martinich, "Invariance Properties of Some
Nonlinear Programs with Stochastic Objectives," March 1985.
P.C. Jones, "Even More with the Lemke Complementarity
Algorithm," January 1985.
Systems Optimization Laboratory
Department of Operations Research
Stanford University
Stanford, California 94305
R.N. Kaul and S. Kaur, "Generalized Convex Functions 
Properties, Optimality and Duality," SOL. 844.
F. Chadee, "Direct Secant Updates of Sparse Matrix Factors,"
SOL. 845.
G. B. Dantzig, "Economic Growth and Dynamic Equilibrium,"
SOL. 848.
J.C. Stone, "Sequential Optimization and Complementarity
Techniques for Computing Economic Equilibria," SOL. 849.
P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright,
"Software and its Relationship to Methods," SOL. 8410.
P.E. Gill, W. Murray, M.A. Saunders, G.W. Stewart and M.H.
Wright, "Properties of a Representation of a Basis for the Null
Space," SOL. 851.
P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright,
"Model Building and Practical Aspects of Nonlinear
Programming," SOL. 852.
D.M. Scott, "A Dynamic Programming Approach to Time
Staged Convex Programs," SOL. 853.
S. Granville and M.V.F. Pereira, "Analysis of the Linearized
Power Flow Model in Benders Decomposition." SOL. 854.
I.J. Lustig, "A Practical Approach to Karmarkar's Algorithm,"
SOL. 855.
G.B. Dantzig, "Deriving a Utility Function for the Economy,"
SOL. 856.
G.B. Dantzig, "Impact of Linear Programming on Computer
Development," SOL. 857.
F. F. Chadee, "Sparse QuasiNewton Methods and the Con
tinuation Problem," SOL. 858.
P. Yang, "Computation of Reliability and Shortage
Distributions in Stochastic Transportation Networks with Cycles,"
SOL. 859.
JOURNALS & STUDIES
Vol. 33, No. 1
G. Cornuejols, J. Fonlupt, and D. Naddef, "The Traveling
Salesman Problem on a Graph and Some Related Integer
Polyhedra."
M. Grotschel, M. Junger, and G. Reinelt, "On the Acyclic
Subgraph Polytope."
M. Grotschel, M. Junger, and G. Reinelt, "Facets of the Linear
Ordering Polytope."
D. F. Shanno, "Globally Convergent Conjugate Gradient
Algorithms."
P.J.M. van Laarhoven, "Parallel Variable Metric Algorithms
for Unconstrained Optimization."
O. Marcotte, "The Cutting Stock Problem and Integer
Rounding."
F.A. Lootsma, "Performance Evaluation of Nonlinear
Optimization Methods via Pairwise Comparison and Fuzzy
Numbers."
P. Dombi, "On Extremal Points of Quasiconvex Functions."
Vol. 33, No. 2
Y. Ikura and G.L. Nemhauser, "Simplex Pivots on the Set
Packing Polytope."
R.M. Freund and J.B. Orlin, "On the Complexity of Four
Polyhedral Set Containment Problems."
K.O. Kortanek and H.M. Strojwas, "On Constraint Sets of
Infinite Linear Programs over Ordered Fields."
J. Goodman, "Newton's Method for Constrained
Optimization."
P.E. Gill, W. Murray, M. A. Saunders, G.W. Stewart and M.H.
Wright, "Properties of a Representation of a Basis for the Null
Space."
D. Goldbarb, "Efficient Dual Simplex Algorithms for the
Assignment Problems."
R. Fourer, "A Simplex Algorithm for PiecewiseLinear
Programming I: Derivation and Proof."
A. Warburton,"Worst Case Analysis of Greedy and Related
Heuristics for Some MinMax Combinatorial Optimization
Problems."
Vol. 33, No. 3
M. Grotschel and 0. Holland, "Solving Matching Problems
with Linear Programming."
J.V. Burke, "Descent Methods for Composite
Nondifferentiable Optimization Problems."
A. Shapiro, "Second Order Sensitivity Analysis and
Asymptotic Theory of Parametrized Nonlinear Programs."
H. Ratschek, "Inclusion Functions and Global Optimisation."
H. Valiaho, "A Unified Approach to OneParametric General
Quadratic Programming."
P. Marcotte, "A New Algorithm for Solving Variational
Inequalities over Polyhedra, with Application to the Traffic
Assignment Problem."
M. Lundy and A. Mees, "Convergence of an Annealing
Algorithm."
K. Malanowski, "Differentiability with Respect to Parameters
of Solutions to Convex Programming Problems."
Vol 34, No. 1
G. Gastou and E.L. Johnson, "Binary Group and Chinese
Postman Polyhedra."
M.J.D. Powell, "How Bad Are the BFGS and DFP Methods
When the Objective Function is Quadratic?"
W. Cook, "On Box and Totally Dual Integral Polyhedra."
C. Blair, "Random Linear Programs with Many Variables
and Few Constraints."
S. Sen and H.D. Sherali, "Facet Inequalities from Simple
Disjunctions in Cutting Plane Theory."
S. Schecter, "Structure of the FirstOrder Solution Set for a
Class of Nonlinear Programs with Parameters."
M. Lundy and A. Mees, "Convergence of an Annealing
Algorithm."
Vol. 34, No. 2
M.L. Balinski, "A Competitive (Dual) Simplex Method for the
Assignment Problem."
P. Marcotte, "Network Design Problem with Congestion
Effects: A Case of Bilevel Programming."
J.B. Rosen and P.M. Pardolas, "Global Minimization of
LargeScale Constrained Concave Quadratic Problems by Separable
Programming."
K.C. Kiwiel, "A Method of Linearizations for Linearly
Constrained Nonconvex Nonsmooth Minimization."
R.H. Mladineo, "An Algorithm for Finding the Global
Maximun of a Multimodal, Multivariate Function."
C.S. Chung, "On the Stability of Invariant Capital Stock in a
TwoDimensional Planning Model."
D. Granot, "A Generalized Linear Production Model: A
Unifying Model."
P. Hansen, B. Jaumard and M. Minoux, "A Linear
ExpectedTime Algorithm for Deriving All Logical Conclusions
Implied by a Set of Boolean Inequalities."
J.M. Borwein and D. Zliu .r;o, "On Fan's Minimax Theorem."
P.W. Smith and H. Wolkowicz, "A Nonlinear Equation for
Linear Programming."
P.C. Jones, "Even More with the Lemke Complementarity
Algorithm."
J.A. Filar and T.A. Schultz, "Nonlinear Programming and
Stationary Strategies in Stochastic Games."
 I~
Founding father or just a footnote?
Founding father or just a footnote?
by David Warsh, Globe Staff
Reprinted by permission from Boston Globe,
Aug. 9, 1985
ast fall, a spate of news
reports appearedinclu
ding one on the front
page of the New York
Times reporting that a
SBell Laboratories scien
tist named Narendra Karmarkar had
discovered a much faster way to
solve difficult planning problems on
computers.
This week in Cambridge, the 28year
old Karmarkar came under mounting fire
from his colleagues at the 12th Interna
tional Symposium on Mathematical Pro
gramming. They snorted at his scientific
manners, scoffed at his claims and derided
his results as being everything from
"frisky" to "majestic." Mostly, they said
that his accounts of superfast solutions to
difficult problems couldn't be replicated.
"My algebra is a lot slower than
Narendra Karmarkar's, I join the general
chorus on that one," said David M. Gay of
Bell Labs. "He may have some wonderful
method after all, but I habitually mistrust
all secret mathematics," said E.M.L.
Beale, a pioneer in the commercial appli
cations of linear programming.
The practical applications of the new
algorithm (a mathematical procedure used
mainly with computers) to a wide range of
problems was thus generally judged to be
still an open question nearly two years
after it was first broached within the pro
gramming community.
Karmarkar's discovery initially had
been billed as a scientific breakthrough of
major proportions, and perhaps a com
mercial coup too. A significantly better
way to make decisions about scheduling
airplanes, mixing ingredients, cutting
steel, routing telephone calls and such
other archetypical tasks of the modem
world could be worth many millions of
dollars.
The news was all the more remarkable
since, on problems with large numbers of
variables, Karmarkar's algorithm was said
to outperform one of the most durable
intellectual attainments of fastmoving
20th century applied math, the simplex
method of linear programming which has
gone hand and glove with computers for
planning purposes since its invention 38
years ago.
Among mathematical programmers 
the scientists who preside over the
marriage of mathematics and computers 
the news of the breakthrough was met
with a certain skepticism. They quickly
went to work examining the method, and
since then, hopes have dimmed, disap
pointment has grown and an undertone of
puzzlement and even rancor directed at Bell
Labs has crept into the discussion.
Increasingly, the verdict has been that
Karmarkar's new approach to linear pro
gramming, while interesting and thought
provoking, may have its roots in old, well
understood alternatives, and that its
applications may be less broad than had
been hoped.
It doesn't help that a similar claim five
years ago by a Russian researcher seemed
promising initially, and turned out to be
theoretically elegant, but didn't pan out as
practical.
Karmarkar himself didn't advance his
cause much in a talk before an unusual
plenary session of the MIT meeting of
some 800 scientists from around the
world. He began by observing that while
mathematicians agree on what constitutes
convincing proof of a mathematical prop
osition, there is no corresponding consen
sus as to what makes a persuasive pre
sentation of experimental results a
contention that was immediately disputed
by many of his listeners.
He again reported having achieved
extremely fast and accurate numerical
solutions 30, 40, 50 times faster than
present methods to complex planning
problems that had been posed to him by
others. But several times when pressed for
details, he refused to supply them, saying
they were proprietary. He declined to give
colleagues copies of the transparencies
that he quickly slipped on and off a
projector to illustrate his 40minute talk.
And at another juncture, as he pre
sented his evidence of the superiority of
his method to the simplex algorithm, he
simply omitted from his table of results
the two hardest problems in a set of 17 
without mentioning what he had done.
His rival colleagues glared in disbelief.
Later, in an interview, Karmarkar said
the disagreements arose because his critics
were simply not doing their homework.
He said he left out the results of the two
sample problems because he felt they were
not commensurable or compatible with
those others were using.
"I think that it is clear from what we
know that we can rule out scientific
fraud," said Jeremy F. Shapiro of MIT,
the program chairman of the symposium,
"but there has been less than full
disclosure."
Michael Garey, Karmarkar's depart
ment head at Bell Labs, said, "Narendra's
like that. It took me three months to get
his viewgraphs from the last talk. He
doesn't pay much attention to building his
political base in this kind of thing."
The situation is complicated by the
possibility that Karmarkar's algorithm is
commercially valuable. IBM Corp. sells a
package of simplex computer codes for
$1,200 a month to hundreds of corporate
customers who use them for everything
from scheduling traveling salesmen to
deciding what timber to harvest.
Karmarkar has frequently claimed that his
method worked 50 times faster than IBM's
on big problems with many variables.
"People have a heavy investment,
both intellectual and financial, in the
simplex method," Karmarkar said.
Despite the criticism, some of
Karmarkar's colleagues stressed that the
scientist's basic insight was sound. The
mere fact that so many sessions at the five
day symposium were devoted to an
examination of his method testified to the
usefulness of the exercise he had triggered,
they said.
"There has been a careful reexami
nation of what it is we have been doing
for 38 years as a result," said Beale.
Some researchers said they still felt
that it was likely that Karmarkar was onto
something important, even if incontro
vertible proof hadn't been furnished.
Michael J. Todd, of Comell University,
for example, said that he thought that
Karmarkar would be vindicated in the long
run: "He knows his stuff better than most:
computer architecture is going to change."
If the controversy surrounding the
Karmarkar algorithm dominated the main
ring, there was much else going on, most
continued on next page
~ ~
of it under the rubric of parallel pro
cessing. Indeed, the commotion all but
eclipsed the news of a remarkable result
by Victor Pan and John Rief, oT the State
University of New York at Albany and
Harvard University, respectively.
They, too, have a new way of achiev
ing dramatic computational speedups.
Theirs involves solving large numbers of
linear equations by computer one that
promises application to a wide range of
practical problems as quickly as parallel
processing computers can be built.
"These are very exciting times," said
George Dantzig, the inventor of the
simplex method, who was honored at a
dinner celebrating his 70th birthday. "I'd
swap 20 years with any of you for the
chance to compete in the present
situation."
New EditorinChief From page 1
subject matter." As such, it may treat an
area that is not emphasized by the Journal,
thereby broadening the horizon of our
regular readers. But the Studies are
hampered by a limited distribution and an
irregular appearance. The current supply
of material for the Studies motivates a
discussion of various possibilities to
strengthen the identity of the series. This
is very much an open issue. The
Publications Committee welcomes any
ideas and suggestions on this matter.
The other editors serving the Journal
and the Studies, including the Senior
Editors, will be invited to resign in
deference to the new EditorinChief; all
new appointments are in his domain.
This is to some extent a formality, and
Dick and Mike will work out a smooth
transition.
The number of free reprints of papers
provided to authors will be increased from
thirty to fifty, at a moderate cost for the
Society.
With respect to the COAL Newsletter,
Jan Telgen resigned as Editor. Robert R.
Meyer will continue in this position and
seek a European CoEditor. Since the
Bonn Symposium, Optima has been
expanded by the useful addition of book
reviews. On the other hand, it remains
difficult for Don Hearn and Achim
Bachem to obtain feature articles (in spite
of the financial rewards that await
prospective authors).
The new Publication Committee will
consist of Richard W. Cottle, Martin
Grotschel, and Jan Karel Lenstra
(chairman). We would be pleased to
receive your comments on the
publications of the Society.
Jan Karel Lenstra
Application for Membership
Mail to:
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Dues for 1986 include subscription to Vol. 3436 of
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i_ ~ _~
Narendra Karmarkar (AT&Y Bell Labs) received the ORSA
Lanchester Prize at the November ORSA/TIMS meeting at Atlanta...The
November SIAM Newsletter features three articles on the Karmarkar
algorithm and its reception...The ISI announces Competitions for
Young Statisticians in Developing Countries. Contact the ISI at 428
Prinses Beatrixlaan, 2270 AZ Voorburg, The Netherlands...The Third
SIAM Conference on Discrete Mathematics will be held May 1416,
1986 at Clemson University. Contact SIAM Coordinator, 117 South
17th St., 14th Floor, Philadelphia, PA 191035052...George
Dantzig was a recipient of the 1985 Harvey Prize of the Technion at
the Institute's Haifa, Israel, campus on June 19...George
Nemhauser has been named the Leon C. Welch Professor of
Engineering at Cornell. He is currently visiting Ga. Tech. as Chandler
Professor...Bobby Schnabel and Paul Boggs (NBS) have been
organizing a SIAM Activity Group (SIAG) on Optimization. Contact
Schnabel at Computer Science Department, University of Colorado,
Boulder, Colorado 80309...Mike Grigoriadis reports that 90% of
MPS dues are allocated to publications. This information is required by
those charging memberships to Federal grants...An International
Conference on Systems Science will be held in Poland, Sept. 1619,
1986. Contact Jerzy Swiatek, Technical Univ. of Wroclaw,
Janiszewskiego St. 11/17, 50370 Wroclaw, Poland.
Deadline for the next OPTIMA is April 1, 1986.
Results of the MPS Election
The MPS Secretariat (at the Interna
tional Statistical Institute) has reported the
results of the recent Council elections.
Noting the special features of the terms of
office of chairman, vicechairman, and
treasurer, the incoming Council will be:
Aug. 1985 Aug. 1986: Chairman,
Alex Orden, ViceChairman, Michel
Balinski; Aug. 1986 Aug. 1988: Chair
man, Michel Balinski, ViceChairman,
Alex Orden; Treasurer, Al Williams,
whose present term of office runs until
1986. Elected to another term: Aug.
1986 Aug. 1989.
Aug. 1985 Aug. 1988: Membersat
Large: Martin Grotschel, Masao Iri, Karla
Hoffman, Robert Schnabel.
Books for review should be sent to the
Book Review Editor, Prof. Dr. Achim
Bachem, Mathematiches Institute der
Universitat zu Koln, Weyertal 8690,
D5000 Koln, West Germany.
Journal contents are subject to change
by the publisher.
Donald W. Heam, Editor
Achim Bachem, Associate Editor
Published by the Mathematical
Programming Society and
Publication Services of the
College of Engineering,
University of Florida.
Composition by Lessie McKoy,
Graphics by Lise Drake.
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