BP TI M A
Mathematical Programming Society Newsletter
JUN2003
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JUNE 2003
The Strong Perfect
Graph Theorem
Girard Cornu6jols *
March 31, 2003
1. Introduction
In this note, all graphs are simple (no loops or
multiple edges) and finite. The vertex set ofgraph
G is denoted by V(G) and its edge set by E(G). A
stable set is a set of vertices no two of which are
adjacent. A clique is a set of vertices every pair of
which are adjacent. The cardinality of a largest
clique in graph G is denoted by E.y(G. The
cardinality of a largest stable set is denoted by
a(G). A kcoloring is a partition of the vertices
into k stable sets (these stable sets are called color
classes). The chromatic number x(G) is the smallest
value of k for which there exists a kcoloring.
Obviously, to(G) K (G) since the vertices of a
clique must be in distinct color classes of a k
coloring. An induced subgraph of G is a graph
with vertex set S c V(G) and edge set comprising
all the edges of G with both ends in S. It is
denoted by G(S). The graph G(V (G) S) is
denoted by G \S. A graph G is perfect if
oe(H) = X(H) for every induced subgraph H of G.
A graph is minimally imperfect if it is not perfect
but all its proper induced subgraphs are.
A hole is the graph induced by a chordless cycle
of length at least 4. A hole is odd if it contains an
odd n umber of vertices. Odd holes are not perfect
since their chromatic number is 3 whereas the size
of their largest clique is 2. It is easy to check that
odd holes are minimally imperfect. The
complement of a graph G is the graph G with
the same vertex set as G, and uv is an edge of G if
and only if it is not an edge of G. It is easy to
check that complements of odd holes are also
minimally imperfect. In the early sixties Berge [1]
proposed the Strong Petrfct Graph Conjecture:
The odd holes and their complements are the only
minimally imperfect graphs. This conjecture
attracted much attention over the last forty years.
It was proved in May 2002 by Chudnovsky,
Robertson, Seymour and Thomas [9] in a very
impressive paper. Claude Berge passed away in
June 2002 knowing that his famous conjecture is
true.
Theorem 1.1 (Strong Perfect Graph Theorem)
(Chudnovsky, Robertson, Seymour and Thomas
19]) The only minimally imperfect graphs are the
odd holes and their complements.
In this note, we survey key aspects of the proof
of the Strong Perfect Graph Theorem. A Berge
graph is a graph that does not contain an odd hole
or its complement as an induced subgraph.
Clearly, every perfect graph is a Berge graph. The
Strong Perfect Graph Theorem stares that the
converse is also true: Every Berge graph is perfect.
The idea of the proof is to show that every Berge
graph either falls into one of four basic classes of
perfect graphs, or that it has a kind of separation
that cannot occur in a minimally imperfect graph.
In [1], Berge also made a weaker conjecture,
which states that a graph G is perfect if and only if
its complement G is perfect. This conjecture was
proved by Lovisz [24] in 1972. We give a short
elegant proof due to Gasparyan [21i.
Theorem 1.2 (Perfect Graph Theorem)
(Lavoisz [24]) Graph G is perfiet if andt only if
graph ( is perfect.
Proof.: Lovisz [25] proved the following stronger
result.
Claim 1: A graph G is perfect if and only if,
for every induced subgraph H, the number of
vertices of H is at most a(H )r(H).
Since a(H) = w (l) and o(H) = Ia(.'), Claim I
implies Theorem 1.2.
Proof of Claim 1: First assume that G is
perfect. Then, for every induced subgraph IH,
o(/) = x(H). Since the number of vertices of H is
at most X(Hn)x(H), the inequality follows.
We give a proof ol the converse due to
Gasparyan [21]. Assume that G is not perfect. Let
H be a minimally imperfect subgraph of G and let
n be the number of vertices of H. Let a = a(H)
and o = to(H) Then H satisfies
and t0 = o(:\S) for every stable set S c V(H).
Let A) be an astable set of H. Fix an (ocoloring
of each of the a graphs IMs for s E A,
let A,, ... A, be the stable sets occurring as a
color class in one of these colorings and let
A := A,, A, ..., A,. Let A be the
corresponding stable set versus vertex incidence
'GSIA, Carnegie Mellon University, Schenley Park, Pittsburgh, PA 15213, USA. gc0v@andrew.cmu.edu
This work was supported in parr by NSF grant DMI0098427 and ONR grant N000 149710196.
PAtW 2
10 P I M 7 0
m JUNE 2003 i,.w 3
r~aLiraiLIna1BL
matrix. Define := iB,,, B,, ..., B,,,]) where B is
an coclique of H \ A. Let B be the corresponding
clique versus vertex incidence matrix.
Claim 2: Every aclique of 1 inrersects all but
one of the stable sets illn '.
Proof of C/aim 2: Let S,, ..... be any
utcoloring of H \ v Since any onclique C of H
has at most one vertex in each S, C intersects all
S's if v' C and all but one if t' eC. Since Chas
at most one vertex in A Claim 2 follows.
int particular, it follows that AB = / I where /
is the matrix filled with ones and I the identity.
Since I is nonsingular, A and B have at least as
mIany columns as rows, that is In > au + 1. This
completes the proof of Claim 1. 0
2. Four Basic Classes of Perfect
Graphs
Bipartite graphs are perfect since, for any induced
subgraph H, the bipartition implies that x(Hl) < 2
and therefore ()(/H) = Z/{).
A graph L is the line graph of a graph G if
V (L) = E() and two vertices of I are adjacent if
and only if the corresponding edges of G are
adjacent.
Proposition 2.1 Line t .,*'. of bipartite giaphs
are perfect.
Proof. If G is bipartite, X'(G) = A(C) by a theorem
of Kinig [231, where X' denotes the edge
chromatic number and A the largest vertex degree.
If L is the line graph of a bipartite graph G,
then x(L) = x'(6) and t (L) = A(G). Therefore x(L)
t(L). Since induced subgraphs of L are also line
graphs of bipartite graphs, the result follows. E
Since bipartite graphs and line graphs of
bipartite graphs are perfect, it follows from
Lov:isz's perfect graph theorem (Theorem 1.2)
that the complements of bipartite graphs and of
line graphs of bipartite graphs are perfect. This can
also be verified directly, without using the perfect
graph theorem. To summarize, in this section we
have introduced four classes of perfect graphs:
bipartite graphs and their complements, and
line graphs of bipartite graphs and their
complements.
These graphs are called basic.
3. 2Join, Homogeneous Pair and
Skew Partition
2Join
A graph G has a 2join if its vertices can be
partitioned into sets V, and V,, each of cardinality
at least there, with nonempty disjoint subsets
A,, B, Q VI and A,, c V,, such that all the
verrices of A1 are adjacent to all the vertices ofA,,
all the vertices of/, arc adjacent to all the vertices
of B, and these are the only adjacencies between
V and V,. 2joins were introduced by CornLijols
and Culnningham [17] in 1985. 'hey gave anl
oC( V(G) E(G)I ) algorithm to find
whether a graph G has a 2join.
When CG contains a 2join, we can decompose
G into two blocks (', and G2 defined as follows.
Definition 3.1 if'A, and B ar in ,/.f ,*'i
conseciteId cmponents of G(Vj,), define block GC to
be ((VI Ui{p,,q1 ), wuher p, E A, and q, E B.,
Otherwise, h't PI he a shortest path from A, to B,
and define block G, to be G(V, U V (P,)). Block
G, is defined similar.
Theorem 3.2 (2Join Decomposition
Theorem) (Cornudjols and Ctunninghamn [171)
Gnp/h G is per/;ec ift nld only fits blocks G and
(G, are pie iect.
Corollary 3.3 /f/a '. .. '. imfperf'ct graph (G
has a 2join, then G is an odd hole.
Proof: Since G is nor perfct, Theorem 3.2 implies
that block G, or G, is not perfect, say G,. Since
G1 is an induced subgraph of G and G is
minimally imperfect, it follows that G = G. Thus,
since I1V, > 3, V, induces a chordless path P,.
Therefore G is a minimally imperfect graph with a
vertex of degree 2. It is well known that such a
graph G is an odd hole 127]. O
Homogeneous Pair
The notion of homogeneous pair was
introduced by Chvdital and Sbihi 15]. A graph G
has a homogeneous pair if V (G) can be partitioned
into subsets A,, A, and B, such that:
A,I + IAI > 3 and II > 2.
If a vertex of B is adjacent to a vertex ofA,
then it is adjacent to all the vertices of A,, for
i {1, 2}.
Theorem 3.4 (Homogeneous Pair Theorem)
(ChvAtal and Sbihi [5]) No minimally imperfect
graph has a homogeneous pair
Skew Partition
A graph G has a skew partition if its vertices can
be partitioned into four nonempty sets A, B, C, D
such that there are all the possible edges between /
and B and no edges fiom C to D. Chv:ital [31
introduced skew partitions in 1985 alnd he
conjectured that no minimally imperfect graph has
a skew partition. IHe observed that the conjecture
holds for a star cltsct, defined to be a skew
partition whereI A I = 1.
Lemma 3.5 (Star Cutset Lemma) (Chv:ital 13])
No minimally imperfect gratiph has a star cutset.
Proof Letr G; be the graph induced by A U B U
Cand (, the graph induced by A U B U D.
The graphs G and G, are perfect. Let i.S he the
color class of atn o)(()coloring of G( that contains
the unique node (ofA, for i (1, 21. Then S
meets all the o)(6)cliques of (r, i.e. o)(G \ (SI U
S,)) < o((d). It follows lhar c'\ ('S U S,) can be
colored with Ifwer than t((6) colors, since it is
perfect. Since S, u) S, is a stable set, G can be
colored with on(C') colors, a contradiction,.
Noteworthy contributions towards the skew
partition conjecture were made by Hoing [22]
and Roussel and Rubio [28]. The conjecture was
setred by ( hu.r1. I., Robertson, Seymour and
Thomas 19]. They obtained it as a consequence oIf
the Strong Perfect Graplh Theorem.
Theorem 3.6 (Skew Partition Theorem)
(Chudtnovsky Robertson, Seymour and Thomas
[91) No iminimoally mperfect graphs has a skew
partition.
In order to prove the Strong perfect Graph
T'heorerm, ( liL u..l 1 Robertson, Seymour and
Thomas first proved the following weaker result.
A skew partition is balanced if
(i) every induced path of length at least 2 in G
with ends in A U B and interior in C UDL
is even, and
(ii) every induced path of length at least 2 in G
with ends in C UD and interior in A UB is
even.
Theorem 3.7 (Clludnovsky, Robertson, Seymour
and Thomas [8]) A minimally imperfect Berge
.,,:t,', with smallesN number of vertices cannot have
a balanced skew partition.
~"~~
 ~II~
fi ll JUNE 2003 ,'. 4
We give the proof of Theorem 3.7. It uses
Lovisz's Replication Lemma [241 which we
discuss next. Incidentally, the Replication Lemma
was the step that Fulkerson missed in his attempt
to prove the Perfect Graph Theorem. Because
Fulkerson had convinced himself thar it was likely
to be false, he had not tried very hard to prove it.
Fulkerson [20) says: "In the Spring of 1971, I
received a postcard fiom Berge saying that he had
just heard that Lovisz had a proof of the perfect
graph conjecture. This immediately rekindled nmy
interest, naturally, and so I sat down at my desk
and thought again about the replication lemma.
Some Four or five hours later, I saw a simple proof
of it."
Lemma 3.8 (Replication Lemma) (Lovmisz [24])
Let G be a pesfict graph and v e V(G). Create a
new vertex v' and join it to v and to all the
neighbors of v. Then the resulting graph G' is
perfect.
Proof it suffices to show X(G') = to(G') since,
for induced subgraphs, the proof follows similarly.
We distinguish two cases.
Case 1: Vertex v is contained in some maximum
clique of G. Then ms(G') = w(G) + 1. This
implies X(G') < to(G'), since at most one
new color is needed in G'. Clearly
X(G') = Co(G') follows.
Case 2: Vertex v is not contained in any
ma;xinmum clique of G. Consider any
coloring of G with co(G) colors and let S be
the color class containing v. Then
m(G \ (S { v 1)) = m(G) 1, since every
maximum clique in G meets S { v }. By the
perfection of G, the graph G \ (S {v)) can
be colored with o(G) 1 colors. Using one
additional color for the vertices (S {v }) U
{v'}, we obtain a coloring of G' with o(SG)
colors. O
Prof/'of Theorem 3.7: Let G be a minimally
imperfect Berge graph with smallest number of
vertices. Suppose that G has a balanced skew
partition A, B, C, D. By the Star Cutset Lemma
3.5, each of A, B, C, D has cardinality at least two.
Let G' be the graph obtained from G by adding a
vertex v adjacent to all the vertices ofA and to no
other vertex of G. If G' contains an odd hole,
then G has an odd path contradicting (i) in die
definition ofa balanced skew partition. Similarly,
if G contains an odd hole, (ii) is contradicted.
Therefore G' is a Berge graph. Now consider
G, = G'\ D and cG = G' \ C. For iE {1, 2}, the
graph G, is perfect since it is Berge and has fewer
vertices than G. Replicare vertex v in GC so that t/
belongs to a clique ofsize s(G). By the
Replication Le.mma 3.8, the resulting graph R,; is
perfect. Consider co(G)colorings of R, and R,
respectively. Both colorings have the same number
of colors in A and assume w.l.o.g. that these colors
are 1, 2, ..., k. Let Kbe the subgraph of G
induced by the vertices with colors 1, 2,... k
and let H be the subgraph of G induced by the
vertices with other colors. Since every o(G)clique
ofG is in G \ D or G \ C, the largest clique in K
has size k and the largest clique in H has size
tc(G) k. The graphs H and Kare perfect since
they are proper subgraphs of G. Color Kwith k
colors and H with io(G) k colors. Now G is
colored with o(G)A colors, a contradiction to the
assumption that G is minimally imperfect. O
Theorem 3.7 was presented in September 2001
at a workshop in Princeton. As the next step
towards Theorern 3.6, Chudnovsky and Seymour
obtained the following theorem in January 2002.
Theorem 3.9 (Chudnovsky and Seymour (10]) A
minimally imperfert Berge graph with smallest
number of vertices cannot have a skew partition.
4. Decomposition of Berge Graphs
Conforri, CornuCjols and Vu.kovic proposed the
Following approach to solving the Strong Perfect
Graph Conjecture.
Conjecture 4.1 (Conforti, Cornuejols and
Vuikovid (2001)) (Decomposition Conjecture)
Every Berge graph G is basic or has a skew
partition, or G or G has a 2join.
Chudnovsky, Robertson, Seymour and Thomas
proved the following variation of this conjecture.
Theorem 4.2 (Chudnovsky, Robertson, Seymour
and Thomas [9]) (Decomposition Theorem)
Every Bege graph G is basic or has a skew
partition or a homogeneous pair, or G or G has a
2join.
This theorem implies the Strong Perfect Graph
Theorem. Indeed, suppose that the
Decomposition Theorem holds and that there
exists a minimally imperfect graph G distinct from
an odd hole or its complement. Choose G with
the smallest number of vertices. G cannot have a
skew partition by Theorem 3.9. G cannot have a
homogeneous pair by Theorem 3.4. Neither G
nor G can have a 2join by Corollary 3.3. Since
G is a Berge graph, G must be basic by the
Decomposition Theorem. Therefore G is perfect, a
contradiction.
Theorem 4.2 was already known to hold in
several special cases. For example, it was known
when G is a Meyniel graph (Burler and Fonlupt
[2] in 1984), when G is clawFree (Chvdtal and
Sbihi [6] in 1988 and Maffray and Reed [26] in
1999), diamondfree (Fonlupt and Zemirline [191
in 1987), bullfree (Chvital and Sbihi 151 in
1987), or dartfree (Chvctal, Fonlupt, Sun and
Zemirline [41 in 2000). All these results involve
special types of skew partitions (such as star
cutsets) and, in some cases, homogeneous pairs
15]. A special case of 2join called augmentation of
a flat edge appears in [26]. In 1999, Conforti and
Cornudjols [131 used more general 2joins to
prove Conjecture 4.1 for WPfree Berge graphs, a
class of graphs that contains all bipartite graphs
and all line graphs of bipartite graphs. [131 was the
precursor of a sequence of decomposition results
involving 2joins. The following result was
obtained in February 2001.
Theorem 4.3 (Conflorti, Cornuejols and
Vuikovic [14]) A squarefire Berge graph is
bipartite, the line graph ofia bipartite gnrph, or has
a 2join or a star cutset.
A breakthrough occurred in September 2001
when Chudnovsky, Robertson, Seymour and
Thomas announced that they could prove the
Decomposition Conjecture in the following
important special case.
Theorem 4.4 (Chudnovsky, Robertson, Seymour
and Thomas [8]) If ( is a Berge graph that
con the li lne graph ofa bipartite subdivision of
a 3connected graph, then G has a balanced skew
partition, or G or 5 has a 2join or is the line
graph of a bipartite graph.
Given two vertex disjoint triangles a,, a a3 and
b,, b,, bA, a subdivided prism is a graph induced by
three chordless paths, P = ,, . b,
1, = a,, ..., b, and P' = a,,. . b at least one
of which has length greater than one, such that P',
P', P' have no common vertices and the only
adjacencies between the vertices of distinct paths
are the edges of the two triangles. The next result,
obtained in January 2002, is a real tourdeforce
and a key step in the proofof the Strong Perfect
Graph Theorem. In particular, it was needed to
prove Theorem 3.9.
* JUNE 2003 ,( 5
Theorem 4.5 (Chudnovsky and Seymour [10]) Ifj
G is a Berge graph that contains a subdivided
prism, then G is the line graph 'ofa bipartite graph
or G has a balanced skew partition or a
homogeneous pain or G or G has a 2join.
A wheel (H, v) consists of a hole H together
with a vertex ,v called the center, with at least
three neighbors in H. If v has k neighbors in H,
the wheel is called a kwheel. A line wheel is a 4
wheel (IH, v) that contains exactly two triangles
and these two triangles have only the center v in
common. A twin wheel is a 3wheel containing
exactly two triangles. A universal wheel is a wheel
(H, v) where the center v is adjacent to all the
vertices of H. A trianglefree wheel is a wheel
containing no triangle. A proper wheel is a wheel
that is not any of the above four types. These
concepts were first introduced in [13]. The
t .II ', theorem, obtained in May 2002,
generalizes an earlier result of Zambelli presented
in September 2001 and of Thomas [291.
Theorem 4.6 (Conlforti, CornuLjols and Zambelli
[16]) IfG is a Berge graph that contains no proper
wheel, subdivided prism or their complements,
then G is basic or has ia skew partition.
The last step in proving the Strong Perfect
Graph Theorem is the following difficult theorem,
also obtained in May 2002.
Theorem 4.7 (Chudnovsky and Seymour I111) If
G is a Berge graph that contains a proper wheel,
but no subdivided prism or its complement, then G
has a skew partition, or G or G has a 2join.
Theorems 4.5, 4.6 and 4.7 imply the
Decomposition T'heorem 4.2, and therefore the
Strong Perfect Graph Theorem. A monumental
paper containing these results is now available [91.
Confoirti, Cornudjols and Vulkovic [151 proved
a weaker version of the Decomposition
Conjecture where "skew partition" is replaced by
"double star cutset". A double star is a vertex set S
that contains two adjacent vertices it, v and a
subset of the vertices adjacent to u or v. Clearly, if
G has a skew partition, then G has a double star
cutset: Take S = A u B. i e A and v e B.
Although the decomposition result in [15] is
weaker than Conjecture 4.1 for Berge graphs, it
holds for a larger class of graphs than Berge
graphs: By changing the decomposition from
"skew partition" to "double star cutset", the result
can be obtained for all oddholefree graphs
instead of just Berge graphs.
Theorem 4.8 (Conforti, Cornudjols and
Vuskovid [ 15]) i/'G is an olddholefiee graph,
then G is a bipartite graph or the line '.. I of a
bipartite graph or the complement of'the line graph
of a bipartite graph, or G has a double star cutset
or a 2join
Theorem 4.8 was used by Cornuejols, Liu and
Vu.kovid [18] to construct a polynomial time
recognition algorithm For perfect graphs.
Independently, Chudnovsky and Seymour [ 12]
found a different algorithm for perfect graph
recognition which does not use decomposition.
Both algorithms 112j, 1181 build on the same
companion paper 17] which performs a certain
"cleaning" step in polynomial time.
A useful tool fri studying Berge graphs is due
to Roussel and Rubio [28]. This lemma was
proved independently by Chudnovsky, Robertson,
Seymour and Thomas [8], who popularized it and
named it The Wondeerfi Le.nnma. It is used
repeatedly in the proofs of'Theorems 4.44.7.
Lemma 4.9 (The Wonderful Lemma) (Roussel
and Rubio 1281) Let G be a Berge graph and
assume that V (G) can be partitioned into it set S
and an odd chordless path P = a, . ', v of
.'.. ,i at least 3 such that u, v are both adjacent to
all the vertices in S and G (S) is connected. Then
one of the '. ',, holds:
(i) An odd number of edges of7P have both ends
adjacent to all the vertices in S.
(ii) P has length 3 and G(S U( {u', v'])
contains an odd cordless path between t'
and v'.
(iii) P has length at least 5 and there exist two
nonadjacent vertices x. x' in S such that
(V(P) \ u, v}) U {x. x'} induces a path.
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ISMP2003 The 18 international Symposium on
Mathematical Programming to be held 18 22 August
in Copenhagen, Denmark, is now rapidly approaching.
We have received 737 abstracts for presentation, and
the number of registered participants is currently 939.
The organizing committee has made a serious effort to
avoid no shows, and currently there are only 2 "non
paid" abstracts. Judging by the contents of the abstracts,
the symposium will as is common For this event be of
high scientific quality.
The presentations will be scheduled in 25 parallel
sessions. The daily schedule starts at 9 with a block of
parallel sessions followed by a plenary lecture. A block of
parallel sessions initiates the afternoon followed by 3
semiplenary lectures. Finally, on Monday August 18 and
Thursday August 21, a block of parallel sessions ends the
day, whereas Thesday 19 and Wednesday 20 are "early
off" days due to the Conference Dinner and the City
Hall reception. The symposium ends Friday August 22
just before lunch with the last of the 17 plenary and
semiplenary lectures.
The full program as well as all other information is
available on the symposium home page:www.ismp2003.dk.
We look forward to welcoming you in Copenhagen.
Jens Clausen
Chairman of the Organizing Committee
IE m~i n A JUNE 2003 wa 7
The 7th International Symposium on
Generalized Convexity/Monotonicity
Hanoi, Vietnam, August 2731, 2002.
The Symposium (GC7) was organized by the
international Working Group on Generalized
Convexity (WGGC) and hosted by the Hanoi
Institute of Mathematics, Vietnam National
Center for Natural Sciences and Technology at
Hanoi, Vietnam during August 2731, 2002. For
the first time the Symposium was held outside
North America and Europe reflecting the growing
research activities in the AsiaPacific region, it was
sponsored by the Pacific Optimization Research
Activity Group as well.
The aim of the Symposium was to provide a
forum for the exchange and dissemination of new
ideas in the field of generalized convexity and
generalized monotonicity and their applications in
optimization, control, stochastics, economics,
management science, finance, engineering and
other related topics. The purpose of the
Symposium was unfilled as GC7 was well
represented by researchers from many parts of the
world. There was a noticeable increase in the
number of participants from the AsiaPacific
region. Collaboration of researchers from various
countries has been an integral part of the research
carried out by members of WGGC, and this was
reflected by the presentations made during the
conference. The sense of being a part of a large
family of researchers with common interests was
special. The credit of new joint works in the near
future would go to the organizers of the
conference and the participants who made this
symposium a great success. The organizers did an
excellent job of arranging a comfortable stay in
Hanoi and providing the facility of using the
library and internet services for all participants
throughout the day.
About fifty presentations, including the invited
talks, were made during the five days of the
Symposium. The invited talks were presented by
R.E.Burkard, Austria (combinatorial
optimization), B.Mordukhovich, USA
(nonsmooth analysis) and H.'liy, Viernam (global
optimization). For program details of GC7 we
refer to the web page at
www.math.ac.vn/conference/gcm7/ and to the
WGGC web page at www.genconv.org. Refereed
Proceedings of the Symposium will appear in a
volume with Kluwer Academic Publishers to be
edited by A.Eberhard, N. Hadjisavvas and
D.T.Luc. Information on the proceedings of the
previous six symposia and on future WGGC
activities is available at www.genconv.org as well.
Apart from ite excellent academic atmosphere the
participants had the opportunity of enjoying
dinner with Vietnamese traditional music,
exploring the city of Hanoi, watching the amazing
puppet show at the Water Puppet Theater and
tasting Vietnamese cuisine at the Banquet with a
firstrate performance of Vietnamese instrumental
and vocal music. Some of the participants stayed
on for a twoday tour to the beautiful spot of Ha
Long Bay near the sea recognized by the U.N. as
a World Heritage Site. It is surrounded by many
small islands of various shapes and is famous for
its amazing limestone cliffs, numerous caves and
unbelievable scenic beauty. This excursion was a
memorable ending of the trip to Vietnam for
many participants of GC7.
Sandor Komlosi, Secretary WGGC
komlosi@ktk.pte.lh
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10PiS T IMA 7901
JUNE .2()011
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in i I'ril,. l , j. l
Constructing
Nontransitive Dice
Robert A. Bosch
April 3, 2003
Two gamblers have decided to use the (unfolded)
dice displayed in Figure I to settle an argument.
They've decided that
they'll each pick one die and roll it once,
gambler 1 will pick first, and
whoever rolls the higher number will win.
Di D2 D2
155 266 355
g 7 8
9 f 8
Figure 1
These dice are nontransitive. If gambler 1 picks
D,, gambler 2 should pick D,. If gambler I picks
D,, gambler 2 should pick D,. And if gambler I
picks D,, gambler 2 should pick D,. No matter
which die gambler I picks, gambler 2 will win
with probability 5/9 (since Prob(D, > D) =
Prrob(D, > )=Prob(D, > D,)= 5/9).
The wellknown statistician Bradley Efron was
the first to design sets of nontransitive dice, and
Martin Gardner was the first to popularize them
(see chapter 22 of 11]).
Problems
Interested readers may enjoy trying to solve the
following problems:
1. Devise an integer programming formulation
or a constraint programming formulation for
constructing nontransitive dice.
2. Use the formulation to find a set of three
nontransitive dice that has the following
properties: (i) each face has a number
between I and 18 on it, (ii) each number in
this range appears on exactly one Face, and
(iii) Prob(D > D)) < Prob(D, > D) <
Prob(D, > D,). Maximize Prob(D > D).
Slither Link Revisited
The previous Mindslharpener was concerned with
slither link puzzles. In a slither link puzzle, the
goal is to find a cycle that consists of horizontal
and vertical line segments and satisfies the puzzle's
adjacency conditions: for each square s and for
every number a, if square s has the number a in it,
then s must be adjacent to precisely a segments of
the cycle. See Figure 2 for an example; see Figure
3 for its solution.
*00******@*
1 1 1 2
0 1 2 1
1* *i 2 * *
** *l *1"2 *2 *2 *
*3* *1* *2* *2*
0 e e e e
Figure 2
It is easy to formulate an IP that can be used to
solve slither link puzzles. In an n x n puzzle, there
are n rows and n columns of squares and n + 1
rows and n + 1 columns of points. We number
the rows and columns of squares from I to n and
the rows and columns of points fiom 0 to n. For
each 0 < i,j < n, we let p. equal 1 if the cycle
visits point (i,j) and 0 if not. For each 0 < i < n
JUNE 2003 A;Ut 9
and (0
3 horizontal line segment connecting points
2 (i, j) and (ij + 1) is a part of the cycle and
0 if not. For each 0 !< i< n  I and
0 5j < n, we let v, equal 1 if the vertical
2 line segment connecting points (i, ,) and
0 (i + 1,j) is a part of the cycle and 0 if not.
For each point (i, ), we need a "degree"
constraint. If (i, j) is an interior point (i.e.,
S 0 < i,j < n) the degree constraint is
9h,, h + + t .l = 2p,,h ,
This constraint says two things: (1) if the
cycle visits point (i, /), then exactly two
line segments of the cycle are incident to (i/, ), and (2) if
the cycle doesn't visit point (i, j), then none of the line
segments of the cycle are incident to (i, j). The degree
constraints for edge points and corner points are similar.
For each square that contains a number, we need another
constraint. If square (i, j) contains the number a, then
we need to include the constraint
h, + v' + + V. a.
This constraint ensures that exactly a segments of the
cycle are adjacent to square (i, j).
Our solution strategy is to minimize the length of the
"rour" subject to all of these constraints. If we find that
there are subrours, we simply add constraints to
eliminate them. (None were needed to produce the
solution displayed in Figure 3.)
References
[ 1 M. Gardner, The Colossal Book of Mathematics,
WW Norton, 2001.
Announcement
The Fourth International Conference on "Frontiers
in Global Optimization" (organized by C. Floudas and
P Pardalos) took place June 812, 2003 in Santorini,
Greece. About 85 active researchers from all over the
world participated in the conference. The conference
focused on deterministic methods for global
optimization, stochastic methods for global
optimization, distributed computing methods in global
optimization, and applications of global optimization in
all branches of applied science and engineering,
computer science, computational chemistry, structural
biology, and bioinformatics. A refereed conference
book with selected papers based on talks presented at
the conference will be published by Kluwer Academic
Publishers later this year.
Conference
Multiscale Optimization
Methods and Applications
February 2628,2004
The Center for Applied Optimization at the University of
Florida, in conjunction with the 2003/2004 Special Year
Mathematics Program, is hosting a conference entitled
"Multiscale Optimization Methods and Applications,"
February 2628, 2004. For information about the
conference, please see the web site: http://www.math.
ufl.edu/speciat03// or contact one of the organizers:
Timothy Davis (davis@cise.ufl.edu)
William Hager (hager@math.ufl.edu)
Panos Pardalos (pardalos@ufl.edu)
I~ ~ 
Figure 3
JUNE 2003
Yinyu Ye. formerly of the University of Iowa,
moved to Stanford University in April 2002
where he is Professor of Management
Science and Engineering and Director of the
MS&E Industrial Affiliates Program.
Professor P. M;. Pardalos, CoDirector of the
Center for Applied Optimization in the
Industrial and.Systems Engineering
Department atthe Univerity of Florida was
elected a Foreign Membe.of the National
Academy of Sciences of tle Ukraine.
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