Title: Optima
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00090046/00071
 Material Information
Title: Optima
Series Title: Optima
Physical Description: Serial
Language: English
Creator: Mathematical Programming Society, University of Florida
Publisher: Mathematical Programming Society, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: March 2004
 Record Information
Bibliographic ID: UF00090046
Volume ID: VID00071
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.


This item has the following downloads:

optima71 ( PDF )

Full Text

ematical Progrmming Society Newsletter
Mathematical Programming Society Newsletter

MARCH 2004



*~ 0

Bi y C r ad i Fos 2 6

MARCH 2004

Binary Clutters and


Bertrand Guenin

Abstract. In this short survey, we consider
three classes of binary clutters, namely ideal
clutters, Mengerian clutters, and cycling
clutters. We will see that these classes are well
understood for clutters of odd circuits of
graphs. This has important applications for
multicommodity flows.

1. Introduction
The purpose of this survey is to present
some results (and extensions) on
multicommodity flows in undirected graphs
which give sufficient conditions for the
existence of fractional, integer, or I-integer
multicommodity flows. These results state
that for certain primal-dual pairs of linear
programs, we can sometimes find optimum
integer solution for either the primal or both
the primal and the dual. It is natural to
consider the same type of primal-dual linear
programs in a more general context. This
leads to the study of special classes of binary
clutters called, ideal, cycling, and Mengerian.
The paper is organized as follows: we first
review definitions on binary clutters. We
state a theorem characterizing Mengerian
binary clutters and state conjectures which
would characterize ideal and cycling binary
clutters. In the following section we focus on
a class of clutters which arises by considering
the families of odd circuits of signed graphs.
We show that we can characterize the ideal,
cycling, and Mengerian properties for that
class of clutters and show the relation with
multicommodity flows. In the last section we
state various generalizations to larger classes
of clutters including a conjecture which
would generalize the 4-colour theorem.

2. Binary Clutters
A clutter H is a finite family of sets, over
some finite ground set E(T), with the
property that no set of H contains, or is
equal to, another set of H. The blocker b(T)
of H is the clutter defined as follows:
E(b(f)) := E(f) and b(H) is the set of
inclusion-wise minimal members of {B : B n
C 0, VC E -}. It is well known that for a
clutter, H, b(b(H))=H A clutter is said to be
binary if, for any C1, C, C E their
symmetric difference C1 A C2 A C3 contains,
or is equal to, a set of H. Equivalently
(Lehman [11]), f is binary if, for every

C E H and B E b(f), C n BI is odd. In the
following discussion we will assume that H is
a binary clutter. Consider the following linear

minimize : eG E(f))
subject to
x(C) > 1 C (P)
x>0 e GE(H)

and its dual

maximize (y : CE iG)
subject to
(yc:e ECEf)" w
Yr > 0

e EE(H) (D)
C E= ft

We say that H is ideal if for all (integer)
w Z+H) there is an optimum integer
solution to (P). This concept is also known
under the name of width-length property,
weak Max Flow Min Cut property or Q+-
MFMCproperty. We say that H is
Mengerian if for all w E Z+(H there is an
optimum integer solution to (P) and to (D).
These clutters also known as clutters with the
(strong) Max Flow Min Cut property or Z+-
MFMC property. We say that weights
w G Z+) are Eulerian if for all pairs D, D' E
b(H) we have that w(D A D') is even (where
A denotes the symmetric difference of two
sets). We say that H is cycling if for all
Eulerian w E Z+ ( there is an optimum
integer solution to (P) and to (D). By
definition every Mengerian clutter is cycling.
Every cycling clutter H is ideal. Indeed
consider a cycling clutter H and any
w E ZE( Since 2w is Eulerian there exist an
optimum integer solution to (P) for 2w. But
that solution is also optimum for w. Thus
Mengerian clutters are cycling and cycling
clutters are ideal. We next show that the
inclusions are all strict.
Let 0Q denote the clutter, where E(4)
correspond to the edges of the complete
graph K, and the elements ofK4 are each of
the triangles of K. Then Q, is not
Mengerian. Indeed suppose w is the vector
of all ones. Then an optimum integer
solution to (P) has value 2 (it corresponds
to a set of edges which intersect all triangles
of K) and an integer solution to (D) has
value 1 (it corresponds to a set of disjoint
triangles). Theorem 3.3 will imply that Q6

Date: December 8, 2003.
Key words and phrases. Signed graphs, multicommodity flows, weakly bipartite graphs, graph colouring.

1 PI I MA 7I1


M- MA RC 20 M

is cycling. Let P, be the clutter whose
ground set correspond to the Petersen graph
and where elements of P, correspond to
the postman sets of the Petersen graph (i.e.
sets of edges which induce a graph whose
odd degree vertices correspond to the odd
degree vertices of the Petersen graph). It can
be readily checked that P, is ideal but not

Not every clutter is ideal. Let 0K denote
the clutter, where E(OK) correspond to the
edges of the complete graph K, and the
elements of K, are each of the odd circuits of
K, (the triangles or the circuits of length five).
Then OK is not ideal. Indeed suppose w is
the vector of all ones. For (P) assign each
variable x the value 1 and for (D) assign each
variable y corresponding to a triangle the
value 1 and all other variables the value zero.
Then x and y are feasible variables for
respectively (P) and (D) which both have
value 10. Hence, there is not optimum integer
solution for (P). Lehman [12] showed that if
a clutter is ideal, then so is its blocker. Since
O is not ideal, neither is its blocker b(OK).
The ground set of the clutter LF are the
elements {1,2,3,4,5,6,7} of the Fano matroid
and the sets in LF are the circuits of length
three (the lines) of the Fano matroid, i.e.,

LF:= {{1,3,5}, {1,4,6}, {2,3,6}, {2,4,5},
{1,2,7}, {3,4,7}, {5,6,7}}.

It can be readily checked that LF is not ideal
either. Note the blocker of LF is LF itself.
Thus Figure 1 describes the world of binary

Ideal 0K5 b(OK)
Mengerian 0 10

Figure 1. The world of binary clutters

Let H be a clutter and i E (). The
contraction f / i and deletion t \ i are
clutters with ground set E(f) {i} where:
f / i is the set of inclusion-wise minimal
members of {S {i} : S E f} and;
f \ i := {S: i S E f}. Contractions and
deletions can be performed sequentially, and
the result does not depend on the order. A
clutter obtained from f by a sequence of

deletions and a sequence of contractions is
called a minor of H. It can be readily checked
that if a clutter is Mengerian (resp. cycling or
ideal) then so are all its minors. Thus we can
attempt to characterize these clutters by
describing the smallest (minor minimal)
clutters not in these classes. This was done by
Seymour [17] for the Mengerian property:

Theorem 2.1. A binary clutter is Mengerian
ifand only it has no Q minor.

A short proof of this result can be found in
[7] (see also [16]). Such a characterization
remains elusive for the class of binary ideal
and cycling clutters. However, the following
excluded minor characterizations are

Idealness conjecture:
A binary clutter is ideal if and only if it has
none of the following minors: LF, O,'

Cycling conjecture:
A binary clutter is cycling if and only if it has
none of the following minors: LF, O,'
b(OK), Po0.

The conjecture on ideal clutters was proposed
by Seymour ([17] p. 200, [18] (9.2), (11.2)).
The definition of cycling is due to Seymour
[18] who was the first to suggest an excluded
minor characterization for this class of

3. Clutters of odd Circuits
A signed graph is a pair (G, *) where G is
an undirected graph and c E(G). We
think of the edges in as having odd length
while the other edges have even length. A set
of edges Uis odd if U Un I is odd and even
otherwise. An edge e is odd if e E and even
otherwise. The set of all odd circuits of
(G, *) forms a clutter, denoted C(G, ),
with ground set E(G). This clutter is binary.
To see this, consider any three odd circuits
C1, C,, C,. Since C,, C, C3 are odd, so is
C A C A C3. Hence, the Eulerian subgraph
C1A C C3 can be decomposed into
circuits which are not all even.

A set *' c E(G) is a signature of (G, *) if
C(G, *) = C(G, '). Consider a signed graph
(G, *) and let 6(U) be a cut of G. Since 6(U)

intersects every cycle with even parity,
* A 6(U) is a signature of (G, *). We call the
operation which consists of replacing by
* A 6(U) a signature-exchange. In a signed
graph (G, *), deleting an edge means
removing it from the graph. Contracting an
edge e means first (if necessary) doing a
signature-exchange so that e is even (i.e. not
in the signature) and then removing the edge
and identifying its ends. A minor of a signed
graph, is any signed graph obtained by a
sequence of deletions, contractions, and
signature exchanges. Note that

C(G, )/e = C((G, *)/e) and
C(G, e)\e= C((G, e)\e).

Thus there is a one-to-one correspondence
between minor operations on a signed graph
and minor operations on the corresponding
clutter of odd circuits.
For n > 3 an odd-K denoted K, is the
signed graph (K, E(K)). A signed graph
(G, *) is said to be strongly bipartite if the
clutter C(G, ) is Mengerian. Observe that
Q, = C(K,). Thus Theorem 2.1 can be
specialized as follows,

Corollary 3.1. A signed graph (G, *) is
strongly bipartite ifand only if it has no K,

A signed graph (G, *) is said to be weakly
bipartite if the clutter C(G, ) is ideal.
Observe that OK= C(I1). It can be readily
checked that b(0K) and L, are not clutters
of odd circuits. Thus the following theorem
of Guenin [5] is a special case of the Idealness

Theorem 3.2. A signed graph is weakly
bipartite ifand only if it has no K minor.

For a short proof of this result see Schrijver
[15]. We will call a set of edges which
intersect all odd circuits of a signed graph
C(G, *) a cover. The elements of b(C(G, ))
are the set of inclusion-wise minimal covers.
It is easy to check that all minimal covers of
C(G, *) are of the form A 6(U) where 6(U)
is a cut. Suppose for all vertices v, w(6({v})) is
even. It implies that w(6(U)) is even for all
cuts 6(U). Now consider, D,D' e b(C(G, *)).
Then D=* A 8(U) and D'=* A (U') and
w(D A D') = w(6(UA U') which is even.
Thus the weights w are Eulerian, as defined
in Section 2. Moreover, it can be readily

MARCH 2004


1 MAR I 20 4 1I4

checked that w are Eulerian exactly when
w(6({v})) is even for all vertices v. A signed
graph(G, *) is said to be evenly bipartite if
the clutter C(G, *) is cycling. Since P10,
b(OK), and Lf are not clutters of odd
circuits, the following theorem of Geelen and
Guenin [2] is a special case of the Cycling
conjecture and a strengthening of Theorem

Theorem 3.3. A signed graph is evenly
bipartite ifand only if it has no K, minor.

Hence, a signed graph is evenly bipartite if
and only if it is weakly bipartite, i.e. the
Mengerian and ideal properties are identical
for the clutters of odd circuits.

4. Multicommodity Flows
The presentation in this section draws
heavily on the paper of Geelen and Guenin
[2]. See also Gerards [3] and Schrijver [16].
We begin by defining the multicommodity
flow problem. We are given an (undirected)
graph G, a subset c E(G), and a function
w E Z+'( An edge d E is called a demand
edge, and w1 is the demand on d. For e GE-
*, we call w, the capacity of e. Let C, be the
set of all circuits C of G such that | C r |= 1.
Thus, if C E C, then there exists a demand
edge d E such that C- {d} is a path
connecting the ends of d. We say that
y E1C is a (G, ,w)-flow if:

(1) For each d E *, e (yc: d E C E C) = wd,
(2) For each e E E- (yc: e E C E C) w.

The first condition asks that the demands are
satisfied, and the second condition asks that
the capacities are not exceeded. A flowy is an
integer flow ify E Zc', andy is a '* r-'.- -
flow if 2y E Zc. A natural condition for the
existence of a flow is that, the demand across
a cut should not exceed its capacity. That is:

Remark 4.1 (Cut-condition). For all
Uc V, w(6(U) -) >w(6(U) n .).

This condition is not sufficient for the
existence of a flow as the following example
illustrates. Consider K, and 6(U) be a cut
with two vertices and three vertices on each
of the shores. Let = E(K,) 6(U). Note I/
can be obtained by signature-exchange

from(K5, *). Consider (K,, *) and suppose
and that all demands and capacities are equal
to one, i.e. w(e) = 1 for all e EE(K). It can
be readily checked that the cut-condition
holds. However, we claim that no flow exists.
Every path included in E(K) joining two
endpoints of a demand edge contains at least
two edges. The total demand is I* I = 4. Thus
the total capacity required for a flow to exists
is at least 4 x 2 = 8. However, IE(K,) I = 6,
a contradiction. In the next statement we
consider the linear programs (P) and (D) of
Section 2 for the clutters H = C(G, *).

Proposition 4.2. Consider a signed graph
(G, *) and w E Z+() such that the cut-
condition holds. If(P) has an optimum
integer solution then there exists a (G, w)-
flow. If in addition (D) has a (1/2) integer
solution then there exists a (1/2) integer flow.

Proof Suppose that the cut-condition is
satisfied. It follows that, for all U c V(G),

w(. A 6(U)) > w(*).

Since all minimal covers are of the form
SA 6(U), it follows that is a minimum
cover. Hence, the characteristic vector of*
is an optimum integer solution to (P). Since
(P) has an optimum solution which is
integer, k is an optimum solution to (P). Let
y E Z+C(') be an optimum solution to (D).
Now, by the complementary slackness
conditions, we see that
(i) for C EC(G, *), ify, > 0 then C n* |
= 1, and
(ii) for each d (i.e. d > 0), (y : e E C
SC(G, )) = w.,

Therefore, the restriction ofy to C, gives a
(G, ,c)-flow (which is integer or 1/2-
integer ify is).

Consider a signed graph (G, *) and
supplies/demands w Z+E() Suppose (G,
*) has no K, minor. Then Corollary 3.1
implies that (G, *) is strongly bipartite. It
follows that (P) and (D) have optimum
integer solutions for the clutters of odd
circuits of (G, ). Hence, by Proposition 4.2
there exists an integer (G, *, w)-flow.
Similarly, we obtain the following corollary of
Proposition 4.2 and Theorem 3.3,

Corollary 4.3. Let (G, *) be a signed graph
with no I minor. Let w E Z+ (F be Eulerian
weights and suppose the cut-condition holds.
Then there exists an integer (G, *, w)-flow.

It is straightforward to verify that each of the
signed graphs given in the next corollary do
not contain a minor. Hence, the following
results are an immediate consequence of
Corollary 4.3.

Corollary 4.4. Let (G, *) be a signed graph
with Eulerian weights w E Z+( which satisfy
the cut-condition. Then there exists an integer
(G, *, w)-flow in the fol/loa'.ig cases:

(i) ifl | = 2,
(ii) if G is planar,
(iii) if A 6(U) is a circuit of length 5 for
some cut 6(U),
(iv) if(G, *) has an even-face embedding on
the Klein bottle.

Case (i) is known as the two commodity flow
theorem, see Hu [9] and also Rothschild and
Whinston [14]. Case (ii) was show by
Seymour [19]. Case (iii) was proved by
Lomonosov [13] for the case where 6(U) = 0
and Gerards (personal communication)
observed that these signed graphs have no K,
minor. Case (iv) was shown by Gerards and
Seb6 [4]. The Klein bottle is obtained from
the 2-sphere by adding two cross-caps. An
even-face embedding is an embedding where
all facial circuits are even. This result is a
generalization of the case of the projective
plane [8].

5. Extentions and Related Problems

We first wish to present two different
generalizations of Theorem 3.2 which are
both special cases of the Idealness conjecture.
For a clutter H and v E E(H), the clutter Y'
has ground set E(T) U {v} and
' = {C U {v} : C E }. Cornudjols and
Guenin [1] showed,

Theorem 5.1. A binary clutter is ideal if it
has none of the ollowa. rg minors: Lf OK,
b(O), Q+, b(Q)*.

It is easy to check that neither Q + nor
b(Q_) + is a clutter of odd circuits. Hence, the
previous theorem implies Theorem 3.2. We

MARCH 2004


M- MA RC 20 M

call a subset of edges of(G, *) an odd .-
if it is an odd st-path; or it is the union of an
even st-path P and an odd circuit C, where P
and C share at most one vertex. It is easy to
verify that clutters of odd st-walks are closed
under taking minors. The family of odd st-
walks form a binary clutter. Guenin [6]

Theorem 5.2. A clutter of odd is
ideal ifand only if it has no LF and no OK

Ifs = t there exist no odd st-paths in (G, *).
Hence, in that case, the clutter of odd st-
walks is the clutter of odd circuits. Since the
clutter Lf is not a clutter of odd circuits, the
previous theorem also implies Theorem 3.2.
The 4-colour theorem [10] states that we
can 4-colour the vertices of any planar graph
(i.e all vertices of G can be coloured one of 4
colours such that adjacent vertices are
assigned different colours). We say that G
contains K, as a minor ifK5 can be obtained

from G by first contracting a subset of edges
and then deleting loops. We say that G
contains K, as an odd minor if K can be
obtained from G by contracting all the edges
of some cut and then deleting loops. Wagner
[20] showed that the 4-colour theorem
implies that we can 4-colour the vertices of
graphs with no K5-minors. Gerards (personal
communication) conjectured the following

Conjecture 5.3. We can 4-colour the vertices
of graphs with no K, as odd minor.

We claim that this is a special case of the
Cycling conjecture. Consider a graph G which
does not contain K, as an odd minor. Then it
can be easily checked that (G,E(G)) has no
J minor, and hence b(C(G, *)) has no
b(OK) minor. If the conjecture holds this
implies that b(C(G, *)) is cycling. Let w be
the vector of all ones. Let C1,C, be odd
circuits. Then w(C, A C) = IC, A CI is even.
It follows that w are Eulerian (according to

the definition of Section 2). Since every odd
circuit of G has length at least three, the
value of the optimum integer solution for
(P), with clutter b(C(G, )), is at least three.
Thus there exists an optimum integer
solution for (D) of value at least three. An
integer solution of (D) corresponds to a
family of disjoint covers B, .... ,B, where
k > 3. For i = 1,2 let H denote the bipartite
graph G \ B and let V, V' be the
corresponding partition of the vertices of H..
Let us label each vertex v of G with the
elements of the Z x Z group as follows: if
v E V n VK then v is labeled 00, if
v E V1' n VK then v is labeled 10, if
v E V n V,' then v is labeled 01, if
v E V1' n V,' then v is labeled 11. We claim
that the colours given by the elements of
Z x Z are a 4-colouring of the vertices of G.
For otherwise we would have an edge uv
where both u, v are in say V, n V2. But since
H, is bipartite, uv E B for i = 1,2, a
contradiction as B1 n B2= 0.


[1] G. Cornudjols and B. Guenin.
Ideal binary clutters,
connectivity and a conjecture
of Seymour. SIAMJ. on
Discrete Mathematics,
15(3):329-352, 2002.
[2] J. Geelen and B. Guenin.
Packing odd-circuits in
Eulerian graphs. ofCombin.
Theory Ser. B, 86(2):280-295,
[3] A.M.H. Gerards. Multi-
commodity flows and
polyhedra. CWI Quarterly,
6(3), 1993.
[4] A.M.H. Gerards and A. Sebo.
Multiflows, the cut-condition,
and Klein's bottle. Under
[5] B. Guenin. A characterization
of weakly bipartite graphs. .
of Combinatorial Theory B,
83:112-168, 2001.
[6] B. Guenin. Integral polyhedra
related to even cycle and even
cut matroids. Math. Oper Res.,
27(4), 2002. 693-710.

[7] B. Guenin. A short proof of
Seymour's Max Flow Min Cut
theorem. of Combin. Theory
Ser. B, 86(2):273-279, 2002.
[8] PD. Seymour H. Okamura.
Multicommodity flows in
planar graphs. of
Combinatorial Theory
B, 31:75-81, 1981.
[9] T. C. Hu. Multicommodity
network flows. Operations
Research, 11:344-360, 1963.
[10] W. Haken K. Apppel. Every
planar map is four colorable.
Bull Amer Soc., 82:711-712,
[11] A. Lehman. A solution of the
Shannon switching game. j
SIAM, 12(4):687-725, 1964.
[12] A. Lehman. On the width-
length inequality.
Mathematical Programming,
17:403-417, 1979.
[13] M.V. Lomonosov.
Combinatorial approaches to
multiflow problems. Discrete
AppL Math., 11, 1985.

[14] B. Rothschild and A.
Whinston. Feasibility of two-
commodity network flows.
Operations Research,
14:1121-1129, 1966.
[15] A. Schrijver. A short proof of
Guenin's characterization of
weakly bipartite graphs. of
Combin. Theory Ser. B,
85:255-260, 2002.
[16] A. Schrijver. Combinatorial
Optimization Polyhedra and
S I Springer-Verlag
Berlin, 2003.
[17] PD. Seymour. The Matroids
with the Max-Flow Min-Cut
property. J Comb. Theory Ser.
B, 23:189-222, 1977.
[18] P.D. Seymour. Matroids and
multicommodity flows.
European of Combinatorics,
pages 257-290, 1981.
[19] PD. Seymour. On odd cuts
and plane multicommodity
flows. Proceedings of the London
' .
3(42):178-192, 1981.
[20] K. Wagner. Ober eine

Eigenschaft der ebenen
Komplexe. Math. Ann.,
114:570-590, 1937.

Department of Combinatorics and
Optimization, University of
Waterloo, Waterloo, ON N2L
3G1, Canada
E-mail address:

MARCH 2004



miihd apyeer

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.


Robert A. Bosch
November 6, 2003

Fill the squares of an n x n grid with black
and white stones, and count the number of
monochromatic squares that are formed.
Note that the arrangement displayed in
Figure 1 gives rise to just one
monochromatic square.

r --
0 @

Figure 1

Interested readers may enjoy trying to
solve instances of the Monochromatic
Squares Problem: Arrange black and white
stones in an n x n grid in such a way that the
number of monochromatic squares is
I found this problem on Al Zimmermann's
"Squares Programming Contest" webpage
sts/Squares/). To win the contest, you'll
need to solve the Monochromatic Squares
Problem for all values of n from 7 to 31. The
grand prize is $500 and the deadline is noon
GMT on January 18, 2004. See
Zimmermann's website for details!

Nontransitive Dice Revisited
The first problem in the previous
installment of Mindsharpener involved
devising an IP or CP formulation that could
be used to find a set ofnontransitive dice.
The second problem was to use the
formulation to find a set of three dice {D ,
D2, D3} with the following properties: (i)
each face has a number between 1 and 18 on
it, (ii) each number in this range appears on
exactly one face, and (iii) Prob(D >- D,) "
Prob(D2 >- D) Prob(D3 >- D2). The
objective was to maximize Prob(D1 > D3).

Alex Meeraus of the GAMS Development
Corporation submitted a very nice IP
formulation based on a formulation devised a
number of years ago by his colleague Paul
van der Eijk. To my knowledge, van der Eijk
was the first to use integer programming to
find sets ofnontransitive dice. His model is a
part of the GAMS Model Library

The Meeraus/van der Eijk model uses
three sets of variables. (I have taken the
liberty of rewriting their model slightly and
using my own notation.) Let x ( equal the
number showing on facefof die d. Let yjf,
equal 1 if face f of die d is greater than face
f' of the die that die dis supposed to beat;
let yjf, equal zero if this is not the case.
(Recall that we'd like die 1 to beat die 3, die
2 to beat die 1, and die 3 to beat die 2.) And
let equal 1 if the number k is assigned to
face fof die d, and 0 if not. The Meeraus/van
der Eijk model is

MARCH 2004

MARCH 2004

maximize C yfif'
f f'

f f' f f'
suj c to2 i

f f" f f"
Xf + 18(1- Yflf') >- xf'3 +1
Xf3 + 17flf 2 Xf
xf + 18(1 Yf2f') > f' +1
xf,1 + 17yf2f' > Xf2
xf +18(1- Yf3f') f'2 +1
xf,2 + 17yf3f, > xf3

X d= kzkfd
SZkfd =1

Vf Vf'

Vf Vf'
Vf Vf'
Vf Vf'


Vf Vd

Zkfd = 1 Vk
f d
Xfd Xd,f-1 +1 Vf Vd >1
all xfd' s intergral; all Yfdf' s and Zkfd' s binary.

The objective (1) is to maximize the
number of ways in which die 1 beats die 3.
By maximizing this, we are maximizing Prob
(D, >- D3). Constraint (2) ensures that the
number of ways in which die 2 beats die 1 is
at least as big as the number of ways in which
die 1 beats die 3. In other words, constraint
(2) makes sure that Prob(D2 >- D) >
Prob(D1 >- D3). Constraint (3) is similar,
guaranteeing that Prob(D3 >- D2) > Prob
(D2 > D1). Together, constraints (2) and (3)
ensure that the set of dice is nontransitive.

Constraints (4) and (5) "define" the yfj. 's.
Note that when Y,,f= 1, Xf, > Xf, and that
when y,., 0,3 > xi,. Similarly,
constraints (6) and (7) define they,2, 's and
constraints (8) and (9) define the Yf3 's.

Constraint (10) establishes the necessary
"links" between -1., '. and the zf's.
Constraint (11) makes sure that each face of
each die receives exactly one number, and
constraint (12) guarantees that each number
appears exactly once. Constraint (13) is used
to reduce the size of the search space.

Alex Meeraus's solution is displayed in Figure
2. Note that Prob(D1 >- D) = Prob(D2 >-
D) = Prob(D3 >- D2) = 7/12. Meeraus
reported that it took CPLEX about 20
seconds to obtain the solution.

~I I -

2 II 1 *


Figure 2

Operations Research

2005 (OR 2005)

Conference on

September 7 9, 2005
University of Bremen,
Bremen, Germany

Prof. Dr. H.-D. Haasis
haasis@ uni-bremen.de

Prof. Dr. H. Kopfer

In Memoriam

Jos Sturm, recently elected as an at-
large member of the MPS Council,
died on December 6, 2003. He
suffered a cerebral hemorrhage on
October 8, 2003, from which he never
recovered. Jos was 32 years old. All
our thoughts and sympathy go to his
wife Changqing, their daughter
Stefanie, and to his family. We deeply
regret the loss of a colleague and dear
friend. Jos' dedication to his work, his
endless enthusiasm, his kindness and
readiness to help will be very much
missed. Jos made important
contributions to the field of
Optimization. He was particularly
known for SeDuMi, an algorithm he
developed for semidefinite
optimization. Jos' life will be
remembered with an article in the
next issue of Optima.

|r T I M A7


MARCH 2004


Prizes awarded by the Mathematical
Programming Society and Society for
Industrial and Applied Mathematics --
"Rewards for outstanding work in afield
promote its quality as well as bring it
deserving publicity. "
Mathematical Programming Society
awards prizes to promote excellence and to
reward achievement in mathematical
programming. Most prizes are awarded at
the triennial symposium of the society. Some
are co-sponsored by other professional
societies, including the Society for Industrial
and Applied Mathematics and the American
Mathematical Society.

2003 Fulkerson Prize Committee
D. Williamson
G. Cornuejols
A. Odlyzko (AMS appointee)

2003 Dantzig Prize Committee
B. Cunningham
L. Wolsey
O. Mangasarian (SIAM member)
R. Fletcher (SIAM member)

2003 Beale-Orchard-Hays Prize
W. Cook
D. Bienstock
N. Gould
J. More

2003 Tucker Prize Committee
R. Burkard
T. McCormick
J. Sturm
L. Trotter

2003 Lagrange Prize in Continuous
Optimization Committee
S. J. Wright
C. T. Kelley
C. Lemarechal
M. Todd

2003 Fulkerson Prize Citation
J. F. Geelen, A. M. H. Gerards, A. Kapoor, "The
Excluded Minors for GF(4)-Representable
Matroids," Journal of Combinatorial Theory
B 79 (2000), 247-299.
Matroid representation theory studies the
question of when a matroid is representable
by the columns of a matrix over some field.
The matroids representable over GF(2) and
GF(3) were characterized by their excluded
minors in the 1950s and the 1970s
respectively. Rota then conjectured that the
matroids representable over any finite field
GF(q) could be characterized in terms of a
finite list of excluded minors. For more than
twenty five years, progress on Rota's
conjecture stalled. The proofs for GF(2) and
GF(3) relied on the uniqueness properties of
representations over these fields, properties
that do not hold for other fields. Thus the
result of Geelen, Gerards and Kapoor came
as a big surprise. The paper of Geelen,
Gerards and Kapoor gives an excluded minor
characterization for matroids represented over
GF(4) by working around the non-
uniqueness of the representation. It has
reawakened interest in the area of matroid
representation and brought renewed hope of
progress towards the solution of Rota's

B. Guenin, "A characterization of weakly
bipartite graphs," Journal of Combinatorial
Theory B 83 (2001), 112-168.
A long-standing area of interest in the field
of discrete optimization is finding conditions
under which a given polyhedron has integer
vertices, so that integer optimization
problems can be solved as linear programs. In
the case of a particular set covering
formulation for the maximum cut problem, a
graph is called weakly bipartite if the
polyhedron has integer vertices for that
graph. Guenin's result gives a precise
characterization of the graphs that are weakly

bipartite in terms of an excluded minor. This
solves the graphical case of a famous
conjecture about ideal binary clutters made
by Seymour in his 1977 Fulkerson Prize-
winning paper. Guenin's proof makes an
ingenious use of a deep theorem of Lehman,
also itself a Fulkerson Prize winner. Guenin's
work has motivated several remarkable
subsequent papers.

S. Iwata, L. Fleischer, and S. Fujishige, "A
combinatorial, strongly polynomial-time
algorithm for minimizing submodular
functions," Journal of the ACM 48 (2001),
761-777, and A. Schrijver, "A combinatorial
algorithm minimizing submodular functions
in strongly polynomial time," Journal of
Combinatorial Theory B 80 (2000), 346-
Submodular functions provide a discrete
analog of convex functions, and submodular
function minimization arises in such diverse
areas as dynamic and submodular flows,
facility location problems, multiterminal
source coding, and graph connectivity
problems. The first polynomial-time
algorithm for submodular function
minimization was given by Gritschel, Lovisz,
and Schrijver in 1981; however, the
algorithm relies on the ellipsoid method,
requires advance knowledge of bounds on the
function values, and is not combinatorial. In
1999, the papers of Iwata, Fleischer, and
Fujishige, and of Schrijver independently
gave combinatorial, strongly polynomial-time
algorithms for this fundamental problem.
These results are a significant step in the
history of combinatorial, strongly
polynomial-time algorithms for discrete
optimization problems, and can be compared
with the Edmonds-Karp algorithm for the
maximum flow problem and Tardos's
algorithm for the minimum-cost flow

1 PI I MA 7I1


MARCH 2004

2003 Dantzig Prize Citation
Jong-Shi Pang is a world leader in the field of
equilibrium programming, variational
inequalities and complementarity problems.
He has made major contributions to the
basic theory and algorithms, and to the
analysis, solution, and unification of many
application problems in these areas. His
many books include the classic, "The Linear
Complementarity Problem," written jointly
with R.W. Cottle and R.E. Stone, which won
the 1994 INFORMS Lanchester Prize.
Pang's numerous papers have helped shape
the careers of many outstanding young
researchers world wide and have attracted
many of them to work in the important field
of mathematical programming. This, coupled
with the breadth and profoundness of his
work, makes Pang eminently deserving of the
Dantzig Prize.

Alexander Schrijver has made deep and
fundamental research contributions to
discrete optimization, including the
applications of the ellipsoid method in
combinatorial optimization, disjoint paths on
surfaces, matrix cones and their applications,
polyhedral and cutting plane theory, and
submodular functions. His landmark book,
"Theory of Linear and Integer
Programming," and his three-volume work,
"Combinatorial Optimization: Polyhedra and
Efficiency," constitute definitive accounts of
the history and present state of discrete
optimization, and will influence researchers
for decades to come. Characterized by
insights that are both broad and deep, and by
a continual pursuit of simplification and
unity, Schrijver's work is scholarship at its

2003 Beale-Orchard-Hays Prize
Elizabeth D. Dolan, Robert Fourer, Jorge J.
Mor6, Todd S. Munson, "Optimization on the
NEOS Server," SIAM News 35 (6), 2002.
The NEOS Server has had a tremendous
impact in the field of optimization, extending
the reach of a wide selection of fundamental
algorithms to a growing number of new
applications areas. The influence of NEOS is
such that in many applied fields the NEOS
Server is synonymous with optimization.
An on-going software project like NEOS
involves the efforts of many people and we

hope the numerous contributors to NEOS
will take pride in sharing this award with the
prize winners.

2003 Tucker Prize Citation
At the XVIII Mathematical Programming
Symposium in Copenhagen the Tucker Prize
for an outstanding paper authored by a
student has been awarded to Tim
Roughgarden, Cornell University, for his
thesis "Selfish Routing".

The other two Tucker Prize finalists chosen
by this year's Tucker Prize Committee
consisting of Rainer Burkard (Chair),
Thomas McCormick, Jos Sturm and Leslie
Trotter are Pablo Parrilo and Jiming Peng.
Tim Roughgarden, who obtained his BS and
MS degrees from Stanford University
completed his Ph.D. thesis in May 2002
under the guidance of Eva Tardos. Currently,
Dr. Roughgarden continues his work at
Cornell University as a postdoc. His thesis
considers the classic problem of designing
traffic networks that lead to good global
routings even when the users are making
local, suboptimal decisions. This touches on
several disciplines such as game theory,
network flows, complexity analysis,
approximation algorithms, and economics.
Roughgarden analyzes the "price of anarchy,"
i.e., the gap between a socially-optimal global
solution and the solution resulting from
selfish users, and finds a variety of tight
results on what is achievable with reasonable
amounts of computation. He further
broadens this out to models of other
situations with selfish users.

Pablo Parrilo obtained his first degrees in
Electronic Engineering from the University
of Buenos Aires. He obtained a Ph.D. in
Control and Dynamical Systems from
California Institute of Technology in June
2000 under the guidance of John Doyle.
Currently, Dr. Parrilo is Assistant Professor of
Analysis and Control Systems at ETH
Z\"urich. Dr. Parrilo was nominated as
Tucker Prize finalist for his paper,
"Semidefinite programming relaxations for
semialgebraic methods." This work
establishes a bridge between convex
optimization and real algebraic geometry,
which opens up a new and promising
research area. The main idea is to explore
conditions under which a function can be

proved to be non-negative by showing that it
is a sum of squares. Parrilo shows
applications of this in diverse areas such as
non-convex quadratic programming, matrix
copositivity, linear algebra, and control

Jiming Peng was born in Hunan Province,
China. He obtained his first degrees in
China. In 1997 Peng moved to Delft
University of Technology where he received
his Ph.D. for his prize winning thesis
entitled, "New Design and Analysis of
Interior-Point Methods". His thesis was
guided by C. Roos and T Terlaky. Peng's
work goes a long way to closing the gap
between the superior theoretical performance
of short-step interior-point methods, and the
superior practical performance of long-step
methods. Peng does this by inventing a new
class of barrier functions called "self-regular"
which allow long-step methods to be
implemented with theoretical time bounds
very close to short-step methods. He applies
this to linear, complementarity, second-order
cone, and semi-definite problems. Currently,
Dr. Peng joined the Department of
Computing and Software, McMaster
University, as an Assistant Professor.

2003 Lagrange Prize in
Continuous Optimization
Adrian Lewis, "Nonsmooth Analysis of
Eigenvalues," Mathematical Programming 84
(1999), pp. 1-24.
Using tools from convex and nonsmooth
analysis, this paper establishes an elegant and
compact chain rule to find the subdifferential
of virtually any function of the spectrum of a
symmetric matrix. It shows that a somewhat
unusual view of symmetric matrices (as being
largely functions of their eigenvalues) is the
key to developing conceptual and technical
tools for optimization over the symmetric
matrices. The paper crowns a series of papers
by Lewis on the analysis of spectral functions.
Like the other papers in this series, it does a
superb job of connecting optimization to
important currents in modern mathematics
and in conveying the spirit of the underlying
mathematics to its optimization audience. It
exposes the highly technical subject matter
forcefully and uncompromisingly, yet is
written in a remarkably lucid and engaging



MARCH 2004


Optimization Prize for Young

Prize for Young Researchers, established in
1998 and administered by the Optimization
Society (OS) within the Institute for
Operations Research and Management
Science (INFORMS), is awarded annually at
the INFORMS Fall National Meeting to one
(or more) young researchers for the most
outstanding paper in optimization that is
submitted to or published in a refereed
professional journal. The prize serves as an
esteemed recognition of promising colleagues
who are at the beginning of their academic
or industrial career.

optimization award includes a cash amount
of US$1,000 and a citation certificate. The
award winners will be invited to give a fifteen
minute presentation of the winning paper at
the Optimization Section Business Meeting
held during the INFORMS Fall National
Meeting in the year the award is made. It is
expected that the winners will be responsible
for the travel expenses to present the paper at
the INFORMS meeting.

ELIGIBILITY: The authors and paper must
satisfy the following three conditions to be
eligible for the prize:

(a) The paper must either be published in a
refereed professional journal no more
than three years before the closing date
of nomination, or be submitted to and
received by a refereed professional
journal no more than three years before
the closing date of nomination.

(b) All authors must have been awarded
their terminal degree within five years of
the closing date of nomination.

(c) The topic of the paper must belong to
the field of optimization in its broadest

NOMINATION: A letter of nomination should
be sent (preferably by email) on or before
this year's closing date of June 1, 2004, to:

Prof Tamis Terlaky
Canada Research Chair in Optimization
Department of Computing and Software
McMaster University
1280 Main Street West Hamilton
Ontario, Canada, L8S 4K1
Phone: +1-905 525-9140 ext. 27780,
FAX: +1-905 524-0340
Email: terlaky@mcmaster.ca

PAST AWARDEES. The past winners of the
Optimization Prize for Young Researchers


Prize Winner
Francois Oustry
Kevin Wayne
Kamal Jain
Sam Burer
Tim Roughgarden

1 PI I MA 7I1


MARCH 2004

ISMP 2003 -
experiences and reflections

The 18 International Symposium on
Mathematical Programming was held in
Copenhagen in August 2003. The
organization committee consisted of 5
persons with me as chair. In the following, I
pass on some of the experiences gained -
hopefully, this will be of value for future
organizers of ISMP.

Number of participants and talks.
Key numbers in connection with large
arrangements as ISMP are: the number of
participants, the number of invited and
contributed talks, and the amount of
sponsorship funding attracted.

The number of participants is interesting
in two aspects: the total number shows the
activity of MPS as a society in general, and
the number of participants singled out on
MPS and non-MPS members, on early
registrations and normal registrations, and on
students, is important from an economical
perspective. The final numbers for ISMP are
indicated in the following table.

MPS Member BF April 30:
MPS Member AF April 30:
MPS Member, Free:
Non MPS members BF April 30:
Non MPS members AF April 30:
Non MPS members, Free:
Student BF April 30:
Student AF April 30:
Student, Free:
Special fee (Danish OR-Society):
Sponsor, Free

In the first budgets for ISMP we estimated
the number of participants to be between
1000 and 1100, however, September 11,
2001 happened in between, and in that
perspective, the number of participants was
seen from the organizers point of view as

The number of invited talks was 7, among
which were 5 plenaries and 12 semi-plenaries.
The scientific papers accompanying the talks
were all delivered in due time that an issue of
Mathematical Programming containing these
were available as conference material the
organizers take the opportunity again to
thank the authors for their effort and
Springer-Verlag for sponsoring the issue.

The number of contributed talks was appr.
730, the appr. indicating that after all a small
number of participants did not show.
However, due to an organizational coupling
between registration of payments and
inclusion of abstract in the abstract booklet
the organizers managed to keep the number
of no-shows to a minimum.

Regarding sponsorship grants, the total
amount was 285.000 DKK. These
contributions came from companies as well
as private foundations and were mainly used
to support participation of young researchers
and researchers from third world countries.
In the table with participant numbers, the
"free" participants correspond to participants
who had their fees waived. In addition, appr.
25 participants received a grant each of 2000
DKK for partial covering of travel expenses.

The grants were given based on an
application procedure with a deadline in
March, 2003. The applications a short
application with a list of 5 publications, and
for students, a recommendation from the
supervisor were processed by the
organization committee. This resulted in a
short turn-around time. We received
substantially more applications than we were
able to grant and some of the sponsorships
came in rather late. Therefore, we kept a
short waiting list for applicants.

Experiences from the organizational task.
Initiated by the MPS chair Bob Bixby, I
had close contact with the MPS executive
committee through monthly telephone
meetings in the last 9 months before MPS.
This was a great help in that way the
organizers were able to drawn upon their
experiences and take into account special
wishes regarding the organization of e.g. the
opening ceremony. Also, the budget and
registration fees were discussed with the
ISMP executives at an early stage.

The local organization committee was kept
at a minimum size, which although giving
each member a considerable amount of work,
in the end resulted in an effective
organization with close collaboration.

We hired a conference agency to deal with
registration, hotels, and a lot of other details.
Also the development and maintenance of
the conference web-page was outsourced to
the agency. The production of the booklet of
abstracts was handled by members of the
organizational committee leading to a
smooth production process and a booklet,
which we are quite proud of.

We decided early that the only way to feed
1000 people in one hour was to supply basic
lunch-bags and include these in the
conference fee. Thereby, we avoided lines for
paying as well as the decision time incurred
when you give people a choice regarding their
meal. This worked well.

The conference site was located 10
kilometers out of Copenhagen centre, and
the participants were to find their way using
public transportation, which was paid for in
the conference fee. After some adjustments
and after we realized that backup
transportation in the morning was a must,
this worked satisfactorily, although not

continue on next page -,



MARCH 2004


June 9-11, 2004

Columbia University

New York City, USA

This meeting, the tenth in the series of IPCO
conferences, 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:
* integer programming
* polyhedral combinatorics
* cutting planes
* branch-and-cut
* lift-and-project
* semidefinite relaxations
* geometry of numbers
* computational complexity
* network flows
* matroids and submodular functions
* 0,1 matrices
* approximation algorithms
* scheduling theory and algorithms

In all these areas, we welcome 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 papers will be presented, in a series of
sequential (non-parallel) sessions. Each
lecture will be thirty minutes long. The
program committee will select the papers to
be presented on the basis of extended
abstracts to be submitted.
The proceedings of the conference will be
published by Springer as a Lecture Notes in
Computer Science volume, and will contain
full texts of all presented papers. Copies will
be provided to all participants at registration

George Nemhauser, Chair
Egon Balas
Daniel Bienstock
Bob Bixby
William Cook
Gerard Cornuejols
William Cunningham
Bert Gerards
Ravi Kannan
William Pulleyblank
Laurence A. Wolsey

Daniel Bienstock,
Peter Fisher
Dept. of IEOR, Columbia University
500 W 120th St.
New York, NY 10027, USA
Phone: 212 854 2942
Fax: 212 854 8103

Summer School
June 7-8, 2004

The IPCO Summer School will take place
June 7 and 8, 2004, and will present the
following speakers:

Joan Feigenbaum, Yale University
Tim Roughgarden, UC Berkeley
Rakesh Vohra, Northwestern University

The summer school will focus on the
interactions between operations research,
computer science, and economics.

continued from pagell
Although sufficient capacity were actually
available if both the highway buses and the
trains were used, we underestimated the
conservativeness of humans once shown a
feasible solution, humans tend to stick to
that rather than finding alternatives.

Pitfalls, tips and tricks
I learned a number of lessons during the
task as organizational chair:

Regarding my own institution, I know
exactly who to speak to about any issue
relevant to conference organization. It
would, however, have been more convenient
to have this knowledge during the

organizational process rather than as a result
of the process. So, if you take on the
obligation to arrange a conference of ISMP-
size, do not believe that everything is laid out
for you even if this is claimed to be the case
from your institution.

You need an "odd-job man" in the team -
if you do not have him you will end up with
the odd jobs (take my word for it).

Be prepared for a number of applications
for invitations from people, who do not want
to attend the conference, but who do want to
be able to enter your country for other

In conclusion
I enjoyed ISMP 2003 and had a number
of good experiences along the organizational
road. I came out as a more knowledgeable
person on any number of important issues as
well as a number of other issues. It was fun,
and I hope the participants enjoyed ISMP
2003 to the same extent as I enjoyed
organizing the symposium.

Jens Clausen

10 PI I M A- iI


MARCH 2004



The inaugural triennial International
Conference on Continuous Optimization
will take place on the campus of Rensselaer
Polytechnic Institute, Troy, New York, August
2-4, 2004; a website for the Conference is
available at:

This is a Mathematical Programming
Society conference. It is a sister conference to
IPCO, the Integer Programming and
Combinatorial Optimization Conference,
and is programmed the year after ISMP the
international symposium on mathematical
It is organized in cooperation with the
INFORMS Optimization Section, the
Society for Industrial and Applied
Mathematics (SIAM) and the SIAM Activity
Group on Optimization.
The scientific program of ICCOPT will
cover all major aspects of continuous
optimization: theory, algorithms,
applications, and related problems. A partial
list of topics includes
linear, nonlinear, and convex
equilibrium programming
semidefinite and conic programming
stochastic programming
complementarity and variational
nonsmooth and variational analysis
nonconvex and global optimization
optimization of partial differential systems
applications in engineering, economics,
finance, statistics, game theory, and
energy modeling and electric power
market modeling
optimization over computing grids
modeling languages and web-based
optimization systems.
The Conference will consist of a mixture
of plenary, semiplenary, invited, and
contributed talks. It is anticipated that at
most four sessions will be scheduled in
parallel. Selected papers will appear in a
special issue of Mathematical Programming
Series B.

A dedicated session will be devoted to
papers by young colleagues, to be chosen by
a panel of reviewers. See the separate Call for
Papers by Young Researchers for details,
including guidelines and submission
information. Naturally, submission of papers
by these researchers to the general conference
is also highly encouraged!

Confirmed plenary and semiplenary speakers
Aharon Ben-Tal
Monique Laurent
Sven Leyffer
Olvi Mangasarian
Carsten Scherer
Alexander Shapiro
Shuzhong Zhang

You are cordially invited to attend the
conference and to submit a contributed
paper for presentation. Due to the limited
number of available slots, the Program
Committee may have to decline some
submissions. Please send one of the Co-
Chairs of the Local Organizing Committee
(Jong-Shi Pang, pangj@rpi.edu or John
Mitchell, mitchj@rpi.edu) a note before
March 1, 2004, indicating if you are
interested in (a) attending the Conference,
and/or (b) contributing a presentation.

To be determined at a later date.

Jong-Shi Pang, Program chair
Roberto Cominetti
Nick Gould
Florian Jarre
Tim Kelley
Masakazu Kojima
Jie Sun
Andre Tits

The Conference will be preceded by a
summer school for graduate students, junior
faculty, and other interested participants,
which will describe some of the recent
exciting developments in continuous
optimization. See separate announcement.

April 5, 2004
Deadline for papers for consideration in
the special session dedicated to young

April 15, 2004
Deadline for titles and abstracts

April 15, 2004
Registration deadline (tentative).

July 31 and August 1, 2004
Summer school.

August 2-4

If you are interested in attending the
Conference, please drop an email to the local
organizers (addresses below).
Details for the Conference will be
continuously updated and posted on the
website http://www.math.rpi.edu/iccopt/ .
We look forward to hearing from you and
to seeing you next summer.

The Local Organizers of ICCOPT I
Jong-Shi Pang pangj@rpi.edu
John Mitchell mitchj@rpi.edu
Co-Chair (SIAM Representative)
Kristin Bennett bennek@rpi.edu
Joe Ecker eckerj@rpi.edu



10 MAR I 2 A 7-E1I



Dr. Jong-Shi Pang has
joined Rensselaer
Polytechnic Institute as
the Margaret A. Darrin
Distinguished Professor
in Applied Mathematics.




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).






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 2004,
including subscription to the journal
Mathematical Programming, are US $80.
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


~ ~

MARCH 2004


F 10 W *AeW VW 40
r~y a




r a

*i8^ --we"ui 4
--1-p .i

..]. .=in~r .i


- a


w- ---I 1

a e


S~ a




a C

- C r C r
* ~~m a

S -"




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


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

Robert Bosch
Dept. of Mathematics
Oberlin College
Oberlin, Ohio 44074 USA
e-mail: bobb@cs.oberlin.edu

Alberto Caprara
DEIS Universita di Bologna,
Viale Risorgimento 2,
I 40136 Bologna, Italy
e-mail: acaprara@deis.unibo.it

Donald W. Hearn

Christina Loosli

University of Florida

Journal contents are subject to change by the

University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs