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Title: Optima
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Publisher: Mathematical Programming Society, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: October 2001
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PTI MA
Mathematical Programming Society Newsletter


OCTOBER2001


Solving the seymour problem
Michael C. Ferris
Gibor Pataki
Stefan Schmieta


IF I I .', [ 'Q 1r pr. .,I-,L m


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OCTOBER 2001


Solving the seymour

problem*


Michael C. Ferrist
Gibor Pataki*
Stefan Schmieta'



July 11, 2001


PAGE 2


Optimization problems are at the heart of much
of operations research and can vary substantially
both in complexity and size. In many problems,
the sheer size of the instance makes it very
difficult to solve due to time or space
limitations. In others, the complexity of the
problem (nonlinearities, nonconvexities, or
discreteness) can make it difficult or impossible
to solve to optimality, even for reasonable sized
instances. This note addresses an instance of the
latter type of problem, arising as a mixed integer
program (MIP) involving discrete variables and
linear functions.


Hard problems in MIPLIB
The MIPLIB library of mixed integer programs
was created in 1992 ([4]) and most recently
updated in 1998 ([5]). Several problems in the
library gained some notoriety, for being among
the toughest. Some of these are:

* The danoint and dano3mip problems that
arise from network design; the latter of these
is unsolved to this date.
The markshare problems, that were created
with particular malice to challenge branch-
and-bound, and cutting plane algorithms.
The seymour problem: a relatively small
setcovering problem with a fascinating origin,
and of remarkable difficulty.

A group of researchers, consisting of the
authors, of Sebastian Ceria at Columbia
University, and Jeff Linderoth at Argonne
National Laboratory has recently succeeded in
solving the seymour problem. In this article, we
describe why we found this problem so alluring,
what experiments we have done, and eventually,
what techniques led us to its solution.


Background on seymour
The seymour problem is a setcovering problem;
i.e. a problem of the form

T
mm e x
st. Ax> e

x e{0,1}"


where e denotes a vector of all ones of
appropriate dimension, and A is a matrix of
zeros and ones. The number of rows in A is
4944 and the number of columns is 1372. The
instance was posed by Paul Seymour, as a by-
product of a new proof of the Four Color
Theorem (FCT) by Neil Robertson, Daniel
Sanders, Paul Seymour, and Robin Thomas [12,
13].
An interesting short history of this problem is
given by these authors at [9] which we
reproduce here verbatim.
The Four Color Problem dates back to 1852
when Francis Guthrie, while trying to color the
map ofcounties ofEngland noticed that four colors
.'/ff, ,' He asked his brother Frederick if it was
true that any map can be colored using four colors
in such a way that adjacent regions (i.e. those
sharing a common boundary segment, not just a
point) receive colors. Frederick Guthrie
then communicated the conjecture to DeMorgan.
The first printed reference is due to Cayley in
1878.
A year later the first proof' by Kempe appeared;
its incorrectness was pointed out by Heawood 11
years later. Another failed proofis due to Tait in
1880; a gap in the argument was pointed out by
Petersen in 1891. Both failedproof did have some
value, though. Kempe discovered what became
known as Kempe chains, and Tait found an


*This material is based on research supported by the National Science Foundation Grants CDA-9726385,
CCR-9972372 and DMS 95-27-124 and the Air Force Office of Scientific Research Grant F49620-01-1-0040
tComputer Sciences Department, University of Wisconsin, 1210 West Dayton Street, Madison, WI 53706
(ferris@cs.wisc.edu)
tDepartment of Operations Research, Smith Building, University of North Carolina, Chapel Hill, NC
27599 (gabor@unc.edu). This research was done while the author was at the Department of IE/OR, at
Columbia University
Axioma Inc. (sschmieta@axiomainc.com). This research was done while the author was at the Department
of IE/OR, at Columbia University


10PTIMA66






OCTOBER 2001


equivalent formulation of the Four Color Theorem
in terms of3-edge-coloring.
The next major contribution came from
.- whose work allowed Franklin in 1922 to
prove that the four color conjecture is true for maps
with at most 25 regions. It was also used by other
mathematicians to make various forms ofprogress
on the four colorproblem. We should specifically
mention Heesch who developed the two main
ingredients needed for the ultimate proof -
reducibility and discharging. While the concept of
reducibility was studied by other researchers as
well it appears that the idea of discharging,
crucial for the unavoidability part of the proof is
due to Heesch, and that it was he who conjectured
that a suitable development of this method would
solve the Four Color Problem. This was confirmed
by Appel and Haken in 1976, when they
published their proof of the Four Color Theorem.
The web page also gives an outline of the
proof of the theorem, and a longer list of
pertinent references.
The seymour IP formulates the problem of
finding the smallest unavoidable set of
configurations that must be "reduced" in order
to prove the FCT. Here "reduced" is a technical
term meaning that the configuration can be
shown not to exist in a minimal
counterexample. Seymour has found a solution
of value 423, but until this work, we are aware
of no one who has been able to reproduce such
a solution. The problem actually has many
solutions of value 423.
The value of its LP relaxation is 403.84.
Therefore, all one must do is raise the lower
bound to say 422.0001 (or to better safeguard
against numerical errors, to say 422.1) to prove
the optimality of Seymour's solution; in fact, for
a long time, we were aiming for 423.0001, as
the best solution we could find was of value
424.
One may question the value of spending
months of research effort trying to solve such
hard IP's, which have no particular "realistic"
application. We can argue though, that it is the
small, and hard problems from which one can
learn the most and the techniques one
develops through their study are very much
applicable to real-world, difficult problems.


First attempts: branch-and-bound
The first attempt to solve seymour was done in
1996 by Greg Astfalk running CPLEX 4.0 with
default settings on an HP SPP2000 with 16
processors, each processor having 180 MHz
frequency, and 720 Mflops peak performance,
for the total of approximately 58 hours,
enumerating about 1,275,000 nodes, and using
approximately 1360 Mbytes of memory. In this
run, CPLEX did not even close 9 units of the
gap; remember that we must close a bit more
than 18.16 units.
We can do quite a bit better, just by using
CPLEX, with the variable selection rule of
strong branching. Strong branching (SB) was
developed by Applegate, Bixby, Chvatal and
Cook in their work on the TSP, and it is an
available setting in several commercial MIP
solvers now. At every node of the branch-and-
bound tree, SB tests several variables as a
candidate to branch on (by partially
reoptimizing on both branches with a limited
number of dual simplex pivots), and picks the
most promising one.
Figure 1 shows what lower bound the CPLEX
6.0 branch-and-bound code has achieved after
enumerating a hundred thousand nodes by
using default branching variable selection vs. SB
variable selection (all other settings were
default). On the horizontal axis one mark means
10 thousand branch-and-bound nodes. The run
with SB closed nearly 9 units of the gap, and
took about a week on a 337 MHz speed
machine.


Cutting
Disjunctive cuts were introduced by Balas in the
seventies [1], then rediscovered from a different
viewpoint in the nineties [11, 14, 2, 3]. They
were termed lift-and-project cuts and
computationally tested in the nineties by Balas,
Ceria and Cornudjols in [2, 3]. For our
experiments, we used the more recent
implementation described in [6].
Here we only give a description of disjunctive
cuts in a nutshell. Given P, the linear
programming relaxation of a 0-1 program, and a
variable x, the inequality ax > P is a 0-1


S PT MA 6


PAGE 3


Figure 1: Strong branching vs.
default branching on seymour


,,,,,



i






OCTOBER 2001


Figure 2: Cutting with 2 options vs.
branch-and-bound with 2 options


PAGE 4


disjunctive cut from the disjunction
x, = 0 v x = 1, if it is valid for both of the sets
P n {x x = 0} and P n {x| x = 1}. From
among all such inequalities, following the
method developed in [2], one generates the cut
which is violated by the current LP optimum x
by the largest possible amount, i.e. P ax is
maximized, with respect to some normalization
constraint. Generating such a cut is done by
solving an LP Disjunctive cuts (as all cuts in
MIP) are added to the LP formulation in
rounds; i.e. one adds a batch of cuts, reoptimizes
the LP, drops all nonbinding cuts, then repeats
the procedure.
According to our experience, it is easy to tell,
whether it is worth applying disjunctive cuts on
a particular IP, just by looking at the result of
two branch-and-bound runs. Disjunctive cuts
work (i.e. adding them significantly raises the
LP lower bound) if (and one can say, only if)
strong branching works! The informal
explanation is that both techniques attempt to
enhance the effect of the branching operation.
Strong branching does this by selecting the best
variable to branch on. Generating disjunctive
cuts from say 50 variables mimics the effect that
can be gained from branching on those variables
(of course, adding these 50 cuts will not result
in a lower bound as good as the one from a 50
level deep branch-and-bound tree).
In our first experiment, we generated 10
rounds of 100 cuts, by selecting the 100
variables that were the most fractional in the
current LP optimum. This run took about 6
hours, and closed about 9.45 units of the gap!
After some experimentation, we produced a
formulation called Formulation 1, that we
thought was worth trying to finish off with
branch-and-bound. The setup was as follows:

* In each cutting iteration we generated cuts
from all the fractional variables; there were
typically about 600 of these.
We sorted the cuts, by putting the one first
from which the euclidean distance of the LP
optimal solution is the largest, and so on.
The distance of the hyperplane {x I ax = }
from the point x is


p-a-
all *
Then, assuming that we have c cuts, we
perform the following step for i = 1, ..., c:
If the cosine of the angle of i cut
hyperplane with any one of the first
i 1 cut hyperplanes is greater than
0.999, we discarded the ih cut.
From the remaining cuts we picked the
first 250, and added them to the LP
formulation.
We call the above method cut selection by
distance. Another method that is quite natural is
called cut selection by usage; we describe this
method next. Suppose again, that we would like
to select the "best" 250 cuts to be added to the
LP formulation. We tentatively add all of them,
then track the course of the reoptimization by
the dual simplex algorithm using the steepest
edge pivot rule. Whenever a cut is pivoted on,
we mark it. We let dual simplex run, until 250
cuts get marked; these will be the selected ones.
We never unmark a cut, and whether a cut is
pivoted on once, or more than once does not
matter. Perhaps surprisingly, we found that out
of more than 600 cuts, each of which is violated
by the current solution, we could never choose
more than about 350 with this method at
most this many are ever pivoted on in the course
of the reoptimization!
Cut selection by distance and by usage
performed quite similarly on the seymour
problem; it would be interesting to see how they
work on other hard IP's, especially, when more
than one type of cut (e.g. knapsack, flow-cover,
Gomory-cuts) is used.
Figure 2 depicts the progress of the cutting
plane algorithm with two different settings
versus branch-and-bound with default
branching and strong branching. The cutting
strategy that worked best was cutting off an LP
solution in the interior of the optimal face, as
opposed to the usual vertex cutting. On the
horizontal axis one mark means 10 thousand
enumerated nodes for branch-and-bound, and
one round of cutting for lift-and-project cuts.
The progress made by branch-and-bound and


10PTIMA66






OCTOBER 2001


cutting planes is quite well comparable this way,
even though the time taken by the four different
algorithms between two tickmarks on the
horizontal axis can be different of course. For
example branch-and-bound with SB took about
a week to enumerate 100 thousand nodes, while
with default branching it took only 3 days. The
most time-consuming was cutting with the
"interior point cutting" option; this took about
2 weeks. Still, in the case of seymour the
question is simply solving it, or not; hence the
few days difference in the running times is
irrelevant in this case. From Figure 2 it is clear
that after 100 thousand nodes, or 10 rounds,
both algorithms completely "ran out of steam";
even after several more months, or years they
would not solve the problem.
Formulation 1 was fed to the CPLEX branch-
and-bound solver, again with the well-tested SB
setting. After about 100 thousand nodes, the
lower bound was further pushed up by about 3
units, for the total of about 15 units; at that
point it was clear, that this way we will never
solve the problem. At the same time, it also
became clear that producing a limited number
of nodes in a branch-and-cut tree, each with at
least 15-16 units of the gap closed, would do the
trick; we would simply need to process those
nodes by branch-and-bound afterwards. Hence
we set up a run to generate the required nodes,
in which a certain number of cutting rounds
was followed by branching for a number of
levels in the branch-and-bound tree, then the
process repeated. Precisely, we

* Generated 10 rounds of cuts at the root
node.
Ran B&B for 4 levels.
At each of the 2 = 16 nodes, we generated 2
more rounds of cuts.
Ran B&B for 4 levels.
At each of the 2 = 256 nodes, we generated
1 more round.

That is, at the end we had 2 = 256 nodes in the
tree, and on the way from the root to any one of
them 13 rounds of cuts were generated. We used
SB for the variable selection; interior-point
cutting, and selecting 250 cuts by usage for cut


PAGE 5


generation. We remark that all cuts generated
within the tree were globally valid, i.e. they were
used at the other nodes as well.
In the end, the gap closed at
the best node was: 16.77
the worst: 15.17
the median: 16.29
We remark that the problem was preprocessed
at the root node by deleting all dominated rows
and columns as usual in setcovering problems.
The reduced problem has 4323 rows, and 882
columns; the IP value of 423 in the original
problem corresponds to the value of 238 in the
preprocessed problem. Although in the parallel
processing of the nodes we had to set the cutoff
values to take into account the preprocessing, we
translated these values back to correspond to the
original instance.


The Condor system
Heterogeneous clusters of workstations are
becoming an important source of computing
resources. One approach to use these clusters of
machines more effectively allows users to run
their (computing intensive) jobs on idle
machines that belong to somebody else. The
Condor system [10, 8] that has been developed
at University of Wisconsin-Madison is one
scheme that manages such resources in a local
intranet setting. It monitors the activity on all
participating machines, placing idle machines in
the Condor pool, that are allocated to service
job requests from users. Users' programs are
allowed to run on any machine in the pool,
regardless of whether the user submitting the job
has an account there or not. The system
guarantees that heavily loaded machines will not
be selected for an application.
Machines enter the pool when they become
idle, and leave when they get busy, e.g. the
machine owner returns. To protect ownership
rights, whenever a machine's owner returns,
Condor immediately interrupts any job running
on that machine, migrating the job to another
idle machine. In fact, the running job is
initially suspended in case the executing
machine becomes idle again within a short
timeout period. If the executing machine


remains busy, then the job is migrated to
another idle workstation in the pool or returned
to the job queue. For a job to be restarted after
migration to another machine a checkpoint file
is generated that allows the exact state of the
process to be re-created. This design feature
ensures the eventual completion of a job. In
order to use the checkpoint feature, the job to
be executed must just be relinked before being
submitted to the Condor manager. An
additional benefit of this relinking is that remote
I/O can be performed on the submitting
machine, therefore limiting the footprint of the
job on the executing machine.
There are various priority orderings used by
Condor for determining which jobs and
machines are matched at any given instance. A
job advertises its requirements via the simple
mechanism of a "job description file". This file
informs Condor of the location of the
executable and the input and output files, along
with the required architecture, operating system
and memory needs of the job. A machine
similarly advertises its properties and a matching
scheme (implemented within the resource
manager) pairs jobs to machines. Based on the
priority orderings, running jobs may sometimes
be preempted to allow higher priority jobs to
run instead. Condor is freely available and has
been used in a wide range of production
environments for more than ten years.
Since the CPLEX suite of optimization
procedures comes in library form, it is very easy
to carry out the relinking of a simple driver
program to run the seymour problem. On June
23, 1999, we submitted two separate CPLEX
6.0 jobs in an attempt to solve Formulation 1
described above. Both were set to run in depth
first search mode to ensure the size of the stored
branch and bound tree did not exceed the
memory of the machines on which it ran. A
cutoff value was set that excludes the presumed
optimal solution by 1. In one job, the
remaining parameters to CPLEX were set to
default values, while in the other job, strong
branching was carried out. At the time of
writing this article, both jobs are still running.
Condor has provided both jobs with over 600
days of CPU time in the ensuing two years.


I P TS i 6






OCTOBER 2001


One of the jobs has explored over 13.4 million
nodes of the tree, while the other has processed
close to 2.5 million nodes. As expected, neither
has found a solution that exceeds the cutoff
value, and furthermore, the lower bound has
remained essentially stagnant for the large part
of this time. Clearly, just applying brute force
execution time to this problem is not going to
solve it. However, it is interesting to note the
reliability of both the Condor and CPLEX
systems to be able to continue executing on a
variety of different machines during this two
year period.
One issue about the above computation is
that each execution is limited to one processor.
While it would be possible to use the parallel
version of CPLEX to increase resources applied
to the solution, it is not at all clear whether the
parallel code would be able to run in the
Condor environment.


Processing the nodes on Condor
Instead, we submitted the 256 IP subproblems
described above, as 256 separate tasks. While the


PAGE 6


efficiencies generated by intercommunication
between these tasks would be lost, the extra
processing available at the root nodes of all the
tasks that was described above was thought to
more than compensate for this loss.
Furthermore, any collection of resources could
be used to solve these 256 instances, involving
state-of-the-art commercial packages.
MPS input files for all 256 subproblems can
be found at [7]. Each problem is listed with the
ID of the corresponding node in the branch-
and-bound tree. The file you get by clicking on
the link will be called "node.mps". The nodes
are sorted by lower bound, which is computed as
the value of the LP relaxation.
These problems were processed using CPLEX
6.6 and XPRESS 11.50. 219 of the problems
were solved using CPLEX via Condor at
Wisconsin. The remainder were processed using
XPRESS 11.50 and CPLEX 6.6 at Columbia.
In general, we used 423.01 as the upper
cutoff for the solvers, since at the outset of this
work we were somewhat skeptical regarding the
existence of a solution of value 423; we did have
one with value 424 though. On July 4, 2000, we


did find a solution with 423, after this the
remaining subproblems were set up with an
upper cutoff of 422.01. In the end, we were able
to generate several solutions of value 423. The
electronic citation [7] gives the binary variables that
take on value 1 in two distinct optimal solutions.
For the 219 jobs that were run under Condor,
the total CPU time used to process them all was
443.6 days, with 41.7 days idle time for jobs
waiting in the Condor queue. During this time
10,244,500 nodes were explored using
3,261,696,402 pivots running on a total of 883
different machines. The longest single node took
36 days to complete, and the shortest completed
in just under 53 minutes. At Columbia, a
further 48.9 CPU days were used to explore
934,868 nodes. The actual elapsed time between
starting the process and ending the process was
37 days, starting in June 2000 and ending on
July 26, 2000.


Acknowledgement

Thanks are due to Oktay Giinliik for helpful
comments on a draft of this paper.


References


[1] E. Balas, Disjunctive
programming, Annals of
Discrete Mathematics 5 (1979)
3-51.
[2] E. Balas, S. Ceria and G.
Cornudjols, A lift-and-project
cutting plane algorithm for
mixed 0-1 programs,
Mathematical Programming 58
(1993) 295-324.
[3] E. Balas, S. Ceria and G.
Cornudjols, Mixed 0-1
Programming by lift-ad-
project in a branch-and-cut
framework, Management
Science 42 (1996) 1229-1246.
[4] R. E. Bixby, E. A. Boyd, and
R. R. Indovina, MIPLIB: A
Test Set of Mixed Integer
Programming Problems,
SIAM News 25 (1992).


[5] R. E. Bixby, S. Ceria, C. M.
McZeal, M. W P. Savelsbergh,
An Updated Mixed Integer
Programming Library:
MIPLIB 3.0, Optima 54
(1998), 12-15.
[6] S. Ceria and G. Pataki,
Solving Integer and
Disjunctive Programs by Lift-
and-Project, Proceedings ofthe
6th Conference on Integer
Programming and
Combinatorial Optimization
(1998) 271-283.
[7] http://firulete.ieor.columbia.
edu/-schmieta/seymour.html,
List of 256 nodal
subproblems.
[8] http://www.cs.wisc.edu/
condor, The Condor Project,
High Throughput
Computing.


[9] http://www.math.gatech.edu/-
thomas/FC/fourcolor.html, The
four color theorem.
[10] M.J. Litzkow, M. Livny, and
M. W Mutka, Condor: A
hunter of idle workstations, In
Proceedings of the 8th
International Conference on
Distributed Computing Systems,
pages 104-111, June 1988.
[11] L. Lovisz and A. Schrijver,
Cones of matrices and set-
functions and 0-1
optimization, SIAMJournal
on Optimization 1 (1991),
166-190.
[12] N. Robertson, D. P. Sanders,
P. D. Seymour and R.
Thomas, The four colour
theorem, J Combin. Theory
Ser. B. 70 (1997), 2-44.


[13] N. Robertson, D. P. Sanders,
P. D. Seymour and R.
Thomas, A new proof of the
four colour theorem, Electron.
Res. Announc. Amer. Math.
Soc. 2 (1996), 17-25
(electronic).
[14] H. Sherali and W Adams, A
hierarchy of relaxations
between the continuous and
convex hull representation for
zero-one programming
problems, SIAM journal on
Discrete Mathematics 3 (1990)
411-430.


10 P T61S






OCTOBER 2001


ISMP 2003: Bridging theory

and practice

ISMP 2003 the 18th International Symposium on Mathematical
Programming will take place in Copenhagen, Denmark on August
18-22, 2003. Being the largest meeting on Mathematical Programming,
the symposium will cover numerous research areas and applications,
illustrating the cross-disciplinarity and creativity which has characterized
this exciting field for more than half a century.
The conference is organized by the Technical University of Denmark
(DTU), in cooperation with the University of Copenhagen. Both
universities have a long tradition in convex and combinatorial
optimization. The Operations Research Section at the Department of
Informatics and Mathematical Modelling (DTU) conducts research in
areas as logistics, transport optimization, vehicle routing, facility location,
production and inventory planning, timetabling and crew scheduling. The
Institute for Mathematical Sciences at the University of Copenhagen has
research activities in econometrics, mathematical finance, numerical
analysis and operations research. Finally, at the Department of Computer
Science, University of Copenhagen, algorithms for solving combinatorial
optimization problems are studied, including cutting/packing problems,
network/VLSI design problems and problems in computational geometry
and biology.
With the bridge to Malmo, opened in the spring 2000, the universities
in Copenhagen have become closely connected to the Swedish
counterparts in Lund and Malmo. This means that the region officially
called the Sound region has one of the worlds largest concentrations of
universities in driving distance from the center of Copenhagen. As
expected, this has caused a considerable synergetic effect to the research
activities, and several international companies have opened research and
development centers in the region to benefit from this concentration of
competence.
The main elements of the ISMP 2003 logo is the Copenhagen-Malmo
bridge and the Petersen Graph. The bridge has been chosen to symbolise
the cross-disciplinarity of ISMP 2003, and also to illustrate the
constructive mixture between theory and practice which is characteristic
of the region. Julius Petersen (1839-1910) was a teacher at the
Polytechnical School in Copenhagen (now DTU) and became later


PAGE 7


professor at the University of Copenhagen. He is considered to be one of
the founders of graph theory which plays an important role in modeling
and solving mathematical programming problems.
Although the symposium should be a sufficient motivation for visiting
Copenhagen, the city has indeed numerous new attractions to offer. In
the harbour area several monumental sites have recently been constructed
or are on the drawing board: The extension of the National Library (also
called the "black diamond"), the forthcoming Opera, the new city area at
Holmen located at an historic navy base and the largest off-shore
wind turbine park. You may explore Copenhagen by free bicycles, which
are available all around the city center. Tivoli, the Little Mermaid and
Nyhavn are some of the well-known sights you will meet on your ride.
Due to the well-established infrastructure, travelling to Copenhagen is
an easy task. From Copenhagen Airport, which is the largest airport in
Scandinavia with several transcontinental departures daily, the city center
is reached by train in just 10 minutes. The Copenhagen Metro will open
by 2002, providing a fast means of transportation in central Copenhagen.

For further information on ISMP 2003 please see the conference website
www.ismp2003.dk, or contact the organizers:

ISMP 2003, c/o Prof. Jens Clausen
Informatics and Mathematical Modelling
Technical University of Denmark, Building 321
DK-2800 Lyngby, Denmark Fax: +45 4588 1397 e-mail: jc@imm.dtu.dk

David Pisinger, Professor, Ph.D.
DIKU, University of Copenhagen
Universitetsparken 1, DK-2100 Copenhagen O
DENMARK
phone:+45 35 32 13 54
fax:+45 35 32 14 01
e-mail: pisinger@diku.dk
http://www.diku.dk/-pisinger


S PT MA 6


~




,f:






OCTOBER 2001


IPCO 2002
Ninth Conference on Integer
Programming and Combinatorial Optimization

May 27-29, 2002
MIT, Cambridge, MA

Announcement and Call for Papers This meeting, the ninth in the series
of IPCO conferences held every year in which no International
Symposium on Mathematical Programming takes place, is a forum for
researchers and practitioners working on various aspects of integer
programming and combinatorial optimization. The aim is to present
recent developments in theory, computation, and applications of integer
programming and combinatorial optimization. Topics include, but are not
limited to: approximation algorithms, branch and bound algorithms,
computational biology, computational complexity, computational
geometry, cutting plane algorithms, Diophantine equations, geometry of
numbers, graph and network algorithms, integer programming, matroids
and submodular functions, on-line algorithms, polyhedral combinatorics,
scheduling theory and algorithms, semidefinite programming. In all these
areas, IPCO welcomes structural and algorithmic results, revealing
computational studies, and novel applications of these techniques to
practical problems. The algorithms studied may be sequential or parallel,
deterministic or randomized. During the three days, approximately thirty-
six papers will be presented, in a series of sequential (non-parallel) sessions.
Each lecture will be thirty minutes long. The conference proceedings will
contain full texts of all presented papers. Copies will be provided to all
participants at registration time.

Summer School A Summer School on Integer Programming and
Combinatorial Optimization shall precede IPCO 2002. Three leading
researchers will present three lectures each, complemented by exercises.
The summer school will take place on May 25-26, 2002 at MIT. PhD
students, postdocs and others interested in participating are encouraged to
pre-register informally by sending e-mail to ipco2002@mit.edu.

Important Dates Deadline for submission of extended abstract: October
29, 2001; notification of acceptance: January 21, 2002; final manuscript
due: February 22, 2002.

Information Further information about the conference and the summer
school is available at http://mit.edu/ipco2002/.

Organizers William J. Cook, Princeton University (program committee
chair), Andreas S. Schulz, Massachusetts Institute of Technology
(organizing committee chair).

Contact Address
Andreas S. Schulz
IPCO 2002
Massachusetts Institute of Technology
Building E53-361
77 Massachusetts Avenue
Cambridge MA 02139-4307 USA
E-mail: ipco2002@mit.edu


PAGE 8


7th International Symposium on
Generalized Convexity/Monotonicity

Hanoi/Vietnam, August 27-31, 2002

Scope Various generalizations of convex functions have been introduced
in areas such as mathematical programming, economics, management
science, engineering and applied sciences. In addition, different kinds of
generalized monotonicity have been proposed, for instance for variational
inequalities and equilibrium problems. Such models are considerably more
adaptable to real-world situations than their convex/monotone
counterpart. A growing literature in this interdisciplinary field has
appeared, including the proceedings of the preceding six international
symposia since the NATO Advanced Study Institute in 1980 in
Vancouver, Canada. The symposium is organized by the international
Working Group on Generalized Convexity (http://genconv.ec.unipi.it). It
is the first symposium in the series that will take place in the Asia-Pacific
region. The conference is sponsored by the Pacific Optimization Research
Activity Group (POP).
Program Committee J.E. Martinez-Legaz (Spain) (co-chair), PH. Sach
(Vietnam) (co-chair), R.Cambini (Italy), J.-P. Crouzeix (France), A.
Eberhard (Australia), N. Hadjisavvas (Greece), S. Komlosi (Hungary),
D.T. Luc (Vietnam, France), S. Schaible (USA).
Organizing Committee N.D. Cong (Vietnam) (co-chair), D.T. Luc
(Vietnam, France) (co-chair), VN. Chau (Vietnam), PH. Dien
(Vietnam), N. Hadjisavvas (Greece), S. Komlosi (Hungary), L.D. Muu
(Vietnam), H.X. Phu (Vietnam), N.X. Tan (Vietnam), N.D. Yen
(Vietnam).
International Advisory Committee The two committees are supported by
an international group of researchers in the field of study, representing in
particular many countries of the Asian-Pacific region.
Invited Speakers J.M. Borwein (Burnaby), R.E. Burkard (Graz), B.
Mordukhovich (Detroit), H. Tuy (Hanoi).
Location. Hanoi Institute of Mathematics, Hanoi, Vietnam.
Language English.
Fee 100 US $
Proceedings Edited proceedings including the invited lectures and a
selection of contributed talks will be published. Co-editors: A. Eberhard,
N. Hadjisavvas, D.T. Luc.
Preliminary Registration Form Please provide your name, title, postal and
e-mail address as well as a tentative title of your proposed contributed talk
(if applicable).
Contact Information
E-mail: gcm7@thevinh.ncst.ac.vn,
dtluc@univ-avignon.fr, jemartinez@selene.uab.es;
Web site: http://genconv.ec.unipi.it;


Mailing address:
Professor Nguyen Dinh Cong,
Institute of Mathematics,
P.O.Box 631,
Boho, 10000 Hanoi, Vietnam;


Professor Dinh The Luc,
Department de Mathematiques,
University d'Avignon,
33 rue Louis Pasteur,
84000 Avignon, France.


10PTIMA66






OCTOBER 2001


FIRST

ANNOUNCEMENT

ICOTA'2001
The 5th International Conference on
Optimization: Techniques and
Applications

December 15-17, 2001, Hong Kong


PAGE 9


THEME AND SCOPE:The 5th International
Conference on Optimization: Techniques and
Applications (ICOTA), jointly organized by The
Chinese University of Hong Kong, The City
University of Hong Kong, and The Hong Kong
Polytechnic University, will be held in Hong
Kong in 2001. It is a continuation of the
ICOTA series, which has had its first four
conferences held in Singapore (1989 and 1992),
Chendu, China (1995), and Perth, Australia
(1998).

The 5th ICOTA represents the first of the
ICOTA series in the new millennium, and has
been given the theme "Optimization for the
New Millennium". The goal of the 5th
ICOTA is to provide an international forum for
scientists, researchers, software developers, and
practitioners to exchange ideas and approaches,
to present research findings and state-of-the-art
solutions, to share experiences on potentials and
limits, and to open new avenues of research and
developments, on all issues and topics related to
optimization.

Plenary Speakers: Shu-Cherng Fang of North
Carolina State University; Masao Fukushima of
Kyoto University; Toshihide Ibaraki of Kyoto
University; David G. Luenberger of Stanford
University; Angelo Miele of Rice University;
Panos M. Pardalos of University of Florida; and
Yinyu Ye of University of Iowa.

Papers on issues related to optimization are
welcome. Topics include (but not limited to)
those in the following tracks:
Optimization theory
Algorithms analysis and design
Applications in industry, service, finance,
business and military.


IMPORTANT DATES:
August 1, 2001 Deadline for paper
submissions
September 1, 2001 Notification of
acceptance
October 1, 2001 Camera-ready due a
late registration fee applies
December 15-17, 2001 ICOTA'01, Hong
Kong


nd


CONTRIBUTED PAPER SUBMISSION
GUIDELINES: Please submit your complete
paper (or an extended abstract) to any of the
three Program Committee Co-Chairs:
Program Co-Chair (North America)
Professor John R. Birge
E-mail: jrbirge@nwu.edu
Program Co-Chair (Asia and Rest of the World)
Professor Duan Li
Email: dli@se.cuhk.edu.hk
Program Co-Chair (Europe)
Professor Henry Wu
E-mail: q.h.wu@liv.ac.uk

GENERAL CONFERENCE CO-CHAIRS:
X.Q. Cai
Chinese University of Hong Kong,
Email: xqcai@se.cuhk.edu.hk

L. Qi
The Hong Kong Polytechnic University,
Email: maqilq@polyu.edu.hk

CONFERENCE WEB SITES:
www.se.cuhk.edu.hk/~icota
www.polyu.edu.hk/~ama/events/conference/icota


S PT MA 6






OCTOBER 2001


m/ai /ener


Two Domino

Problems

Robert A. Bosch
August 27, 2001


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We invite OPTIMA readers to submit solutions to the problems to Robert Bosch
(bobb@cs.oberlin.edu). The most attractive solutions will be presented in a
forthcoming issue.


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Figure 1 displays a complete set of "double nine" dominoes.
Each domino is one sided and can be used horizontally or vertically.


Problems

Interested readers may enjoy trying to solve the
following problems:

1.Use a complete set of double nine dominoes to
construct a replica of the abstract picture
displayed in Figure 2. Note that the 5-7 domino
can be placed (horizontally) in the upper left
corner, but it cannot be made to fit in the lower
left corner.

2.Using three complete sets of double nine
dominoes, construct the "best possible
approximation" of Leonardo DaVinci's Mona
Lisa, as displayed in Figure 3. Incidentally, in
1993 the artist Ken Knowlton created a portrait
of Scientific American's "Mathematical Games"
columnist Martin Gardner using nine complete


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Figure 2

sets of double nine dominoes.
To see the portrait, point your browser to
www.artists-nh.com/knowtton.htm


10PTIMA66


PAGE 10







OCTOBER 2001


Painting by numbers revisited

Benoit Rottembourg reports that Constraint
Programming works well on paint-by-numbers
puzzles:


I'm in charge of the corporate OR department
of Bouygues, a French holding company. We
studied (enjoyably) with two students (F. Pernias
and F Buscaylet) your funny little
mindsharpener published in Optima 65.


Our team has a tradition in Constraint
Programming (we have our own CP solver,
Choco, which is like ILOG's Solver) and
naturally we formulated your problem in a very
straightforward manner in CP, yielding fast
solutions on the dragonfly example. We also
tested some of the most difficult instances on the
web site you mentioned, and in each case we
obtained the solution in a few seconds.


PAGE 11


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i P 111 A661






OCTOBER 2001


Professor in Applied Mathematics

at the Swiss Federal Institute of

TechnologyLausanne (EPFL)



The EPFL plans a substantial expansion in the basic sciences,
including a significant reinforcement in mathematics, physics,
and chemistry, and a major new effort in the life sciences.

As part of this broad program, the Mathematics Department
has an opening at the full professor level. Applications for
appointments at the Associate and Assistant Professor
(tenure-track) levels will also be considered. We seek outstanding
individuals in all areas of applied mathematics.


PAGE 12


ECOLE POLYTECHNIQUE
FEDERAL DE LAUSANNE

Applications in discrete mathematics and statistics are particularly
encouraged. Successful candidates must develop an
independent, internationally recognized program of scholarly
research and must be willing to teach at both the undergraduate
and graduate level. Substantial start-up resources will be provided.
Women candidates are strongly encouraged to apply.

More information about EPFL and its Department of Mathematics at
http://www.epfl.ch and http://dmawww.epfl.ch.

Applications, including CV, publication list, concise statement
of research interests (3 pages or less) and three letters of
reference, should be sent to:

Professor Gerard Ben Arous
Chairman of the Search Committee
Department of Mathematics
Ecole polytechnique fid&rale de Lausanne (EPFL)
CH-1015 Lausanne, Switzerland
Professor in Applied Mathematics
at the Swiss Federal Institute of Technology Lausanne (EPFL)


CALL FOR PAPERS


MATHEMATICAL
PROGRAMMING
Series B

Mathematical Programming in Biology and
Medicine


We invite research articles for a forthcoming
issue of Mathematical Programming, Series B,
on the applications of mathematical
programming methodology and techniques to
the field of biology and medicine. For
example, optimization problems and solutions
in computational biology would be an area of
interest among many others. Also possible are
survey papers that give in-depth introduction to
areas of biology and medicine where the use of
mathematical programming is novel and
promising. Our hope is that this special issue
would bring the two fields a little closer.

Deadline for submission of full papers: February
28, 2002. We aim at completing a first review of
all papers by August 31, 2002.
Electronic submissions in the form of
postscript files are encouraged. All submissions
will be refereed according to the usual standards


of Mathematical Programming. Information
about this issue can be obtained from the guest
editors for this volume or at
www.caam.rice.edu/-yzhang/mpb/

Guest Editors:
Michael C. Ferris
Department of Computer Sciences
University of Wisconsin
1210 West Dayton Street
Madison, WI 53706, USA
Email: ferris@cs.wisc.edu

Yin Zhang
Department of Computational
and Applied Mathematics
Rice University, MS-134
Houston, Texas 77005, USA
Email: yzhang@caam.rice.edu


10PTIMA66






OCTOBER 2001


CALL FOR PAPERS

4th Workshop on Algorithm
Engineering and Experiments

ALENEX 02
January 4-5, 2002, San Francisco, California
Radisson Miyako Hotel


GENERAL INFORMATION:
The aim of the annual ALENEX workshops is to
provide a forum for the presentation of original
research in the implementation and experimental
evaluation of algorithms and data structures. We
invite submissions that present significant case
studies in experimental analysis (such studies
may tighten, extend, or otherwise improve
current theoretical results) or in the
implementation, testing, and evaluation of
algorithms for realistic environments and
scenarios, including specific applied areas
(including databases, networks, operations
research, computational biology and physics,
computational geometry, and the world wide
web) that present unique challenges in their
underlying algorithmic problems. We also invite
submissions that address methodological issues
and standards in the context of empirical
research on algorithms and data structures.
The scientific program will include invited
talks, contributed research papers, and ample
time for discussion and debate of topics in this
rapidly evolving research area. A proceedings
will be published, and a special issue of the
ACM Journal of Experimental Algorithmics will
feature invited contributions from the workshop.
This workshop is colocated with the 12th
Annual ACM-SIAM Symposium on Discrete
Algorithms (SODA02), and will take place in
the two days preceding that conference. A paper
that has been reviewed and accepted for
presentation at SODA is not eligible for
submission to ALENEX. We recognize, however
that some research projects spawn multiple


papers that elaborate on different aspects of the
work and are willing to respond to inquiries
about overlapping papers.
The workshop is supported by SIAM, the
Society for Industrial and Applied Mathematics,
and SIGACT, the ACM Special Interest Group
on Algorithms and Computation Theory.

SUBMISSIONS:
Authors are invited to submit 10-page extended
abstracts by 5:00 PM EDT, October 8, 2001
and must use the SIGACT electronic
submissions server. Detailed instructions for
submitting to the workshop can be found at the
workshop's website.
http://cs.umd.edu/-mount/ALENEX02
Notification of acceptance or rejection will be
sent by November 5, 2001. The deadline for
receipt of papers in final version is December 10,
2001. Presenters must have submitted the final
versions of their papers in order to be able to
present them at the workshop.

PROGRAM COMMITTEE:
Nancy Amato, Texas A&M University
Marshall Bern, Xerox PARC
Michael Goodrich, Johns Hopkins University
Tom McCormick,
University of British Columbia
Michael Mitzenmacher, Harvard University
David Mount, (Co-chair),
University of Maryland
Giri Narasimhan, University of Memphis
Rajeev Raman, University of Leicester
Clifford Stein, (Co-chair), Columbia University


The Computational

Optimization Research

Center at Columbia

University

announces the

1st Columbia

Optimization Day:


"Combinatorial Optimization

and Integer Programming,

The State of the Art"


Wednesday, November 28,2001


Speakers:
Francisco Barahona, IBM Research
Robert Bixby, Rice University and ILOG Cplex
Sebastian Ceria, Axioma
William Cook, Princeton University
Vasek Chvatal, Rutgers University
David Johnson, AT&T Research
George Nemhauser, Georgia Tech
Bruce Shepherd, Lucent

This meeting will take place at Columbia
University, New York
For further information, please visit
www.corc.ieor.columbia.edu/meetings/cl/cl.html


S MA66


PAGE 13






10 OCOE 200 IG 14 6 6


imaufry


H.P(Paul) Williams has moved from

SoLIrhalmpcon UlnivcrsiNY co a Chair of OR

at rhe London School of Economics. His

.new..email is h.p.willianms@lse.ac.uk.
',.''*:..'/,' '* CA4:,! . * ..


Application for Membership


I wish to enroll as a member of the Society.
My subscription is for my personal use and not for the benefit of any library or institution.
O I will pay my membership dues on receipt of your invoice.
O I wish to pay by credit card (Master/Euro or Visa).


CREDIT CARD NO.


EXPIRATION DATE


FAMILYNAME
MAILING ADDRESS


TELEPHONE NO. TELEFAX NO.
EMAIL
SIGNATURE 0


Mail to:
Mathematical Programming Society
3600 University City Sciences Center
Philadelphia PA 19104-2688 USA

Cheques or money orders should be made
payable to The Mathematical Programming
Society, Inc. Dues for 2001, including
subscription to the journal Mathematical
Programming, are US $75.
Student applications: Dues are one-half the
above rate. Have a faculty member verify your
student status and send application with dues
to above address.

Faculty verifying status


Institution


OCTOBER 2001


PAGE 14


%' '" J J'














Springer ad (insert film)





O P T I M A
MATHEMATICAL PROGRAMMING SOCIETY

UNIVERSITY OF

w FLORIDA


Center for Applied Optimization
371 Weil
PO Box 116595
Gainesville FL 32611-6595 USA


FIRST CLASS MAIL


EDITOR:
Jens Clausen
Informatics and Mathematical Modelling,
Technical University of Denmark
Building 305 room 218
DTU, 2800 Lyngby
Tlf: +45 45 25 33 87 (direct)
Fax: +45 45 88 26 73
e-mail: jc@imm.dtu.dk


FOUNDING EDITOR:
Donald W. Hearn
DESIGNER
Christina Loosli
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