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 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: June 1999
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JUNE 1999

L&L' T I M A

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Characterizations
ofPerfect Graphs
by Martin Grotschel

Characterizations

of Perfect Graphs

By Martin Gritschel

T he favorite topics and results of a researcher change over
time, of course. One area that I have always kept an eye on is
that of perfect graphs. These graphs, introduced in the late
'50s and early '60s by Claude Berge, link various mathemati-
cal disciplines in a truly unexpected way: graph theory, combinatorial
optimization, semidefinite programming, polyhedral and convexity theo-
ry, and even information theory.
This is not a survey of perfect graphs. It's just an appetizer. To learn
papers [1] and [2]. The book [3] is a collection of important articles on
perfect graphs. Algorithmic aspects of perfect graphs are treated in [13].
A comprehensive survey of graph classes, including perfect graphs, can be
found in [5]. Hundreds of classes of perfect graphs are known; 96 impor-
tant classes and the inclusion relations among them are described in [16].
So, what is a perfect graph? Complete graphs are perfect; bipartite,
interval, comparability, triangulated, parity, and unimodular graphs are
perfect as well. The following beautiful perfect graph is the line graph of
the complete bipartite graph K3, .
Due to the evolution of the theory, definitions of perfection (and ver-
sions thereof) have changed over time. To keep this article short, I do
not follow the historical development of the notation. I use definitions
that streamline the presentation. Berge defined
G is a perfect graph,
if and only if
(1) m(G) = X(G') for all node-induced subgraphs G' c G,
where ( (G) denotes the clique number of G (= largest cardinality of a
clique of G, i.e., a set of mutually adjacent nodes) and Z(G) is the chro-
matic number of G (= smallest number of colors needed to color the
nodes of G). Berge discovered that all classes of perfect graphs he found
also have the property that
(2) a (G') = X(G) for all node-induced subgraphs G' c G,
where a (G) is the stability number of G (= largest cardinality of a stable
set of G, i.e., a set of mutually nonadjacent nodes) and x(G') denotes the
clique covering number of G (= smallest number of cliques needed to
cover all nodes of G exactly once).
Note that complementation (two nodes are adjacent in the complement G
of a graph G iff they are nonadjacent in G) transforms a clique into a sta-
ble set and a coloring into a clique covering, and vice versa. Hence, the
complement of a perfect graph satisfies (2). This observation and his dis-
covery mentioned above led Berge to conjecture that G is a perfect graph
if and only if
(3) G is a perfect graph.
Developing the antiblocking theory of polyhedra, Fulkerson launched
a massive attack on this conjecture (see [10], [11], and [12]). The conjec-
ture was solved in 1972 by Lovisz [17], who gave two short and elegant
proofs. Lovisz [18], in addition, characterized perfect graphs as those
graphs G = (V E) for which the following holds:
(4) m(G') a(G') > | V(G')| for all node-induced subgraphs G' c G.
A link to geometry can be established as follows. Given a graph G =
(V, E), we associate with G the vector space R where each component
of a vector 1' is indexed by a node of G. With every subset S c Vi we
can associate its incidence vector Zs = (xZ,) ,e R defined by

-- -' ----- -1

s := 1 if v S, x := 0 if v0 S.
The convex hull of all the incidence vectors of stable sets in G is denot-
ed by STAB(G), i.e.,
STAB(G) = conv {xs e R" I S c Vstable}

and is called the stable setpolytope of G. Clearly, a clique and a stable set of
G can have at most one node in common. This observation yields that,
for every clique Q c V the so-called clique inequality

is satisfied by every incidence vector of a stable set. Thus, all clique
inequalities are valid for STAB(G). The polytope
QSTAB(G) := {x e R"| 0 < x, v v V x(Q) <1 V cliques Qc V}
called fractionalstable setpolytope of G, is therefore a polyhedron contain-
ing STAB(G), and trivially,
STAB(G) = conv {x e {0,1}" x eQSTAB(G)}.
Knowing that computing a(G) (and its weighted version) is NP-hard, one
is tempted to look at the LP relaxation
max cx, x e QSTAB(G),
where c e R" is a vector of node weights. However, solving LPs of this
type is also VP-hard for general graphs G (see [14]).
For the class of perfect graphs G, though, these LPs can be solved in
polynomial time albeit via an involved detour (see below).
Let us now look at the following chain of inequalities and equations,
typical for IP/LP approaches to combinatorial problems. Let G = (VE) be
some graph and c > 0 a vector of node weights:

max { c, | Sc Vstable set of G} =
veS
max{cx I x e STAB ()}
max {c x| x> 0, x(Q) lVcliquesQc V, xe {0, 1}) <
max {c7 x x> 0, x(Q) < lVcliquesQc V} =
min{ c ynI yv2 cV vE V, y2 >OVcliquesQc V} <
Clique C v
min{ y IC y >c, ve V, y E ZV cliquesQEc V}
Clique C3 v

The inequalities come from dropping or adding integrality constraints,
the last equation is implied by LP duality. The last program can be inter-
preted as an IP formulation of the weighted clique covering problem. It
follows from (2) that equality holds throughout the whole chain for all
0/1 vectors c iff G is a perfect graph. This, in turn, is equivalent to
(5) The value max {cx I x e QSTAB(G)}
is integral for all c e {0,1} .
Results of Fulkerson [10] and Lovisz [17] imply that (5) is in fact equiva-
lent to
(6) The value max {cx I x e QSTAB(G)} is integral for all ce +.
and that, for perfect graphs, equality holds throughout the above chain for
all c e / This, as a side remark, proves that the constraint system defin-
ing QSTAB(G) in totally dual integral for perfect graphs G. Chvital [6]
observed that (6) holds iff
STAB(G) = QSTAB(G)

These three characterizations of perfect graphs provide the link to poly-
hedral theory (a graph is perfect iff certain polyhedra are identical) and
integer programming (a graph is perfect iff certain LPs have integral solu-
tion values).
Another quite surprising road towards understanding properties of per-
fect graphs was paved by Lovisz [19]. He introduced a new geometric
representation of graphs linking perfectness to convexity and semidefinite
programming.
An orthonormal representation of a graph G = (VE) is a sequence (u, i
e Vj of V vectors u E R such that | u,|= 1 for all i e Vand uT u = 0 for
all pairs i,j of nonadjacent nodes. For any orthonormal representation
(u | i e V) of G and any additional vector c of unit length, the so-called
orthonormal representation constraint

S("I )2 1

is valid for STAB(G). Taking an orthonormal basis B = {e..., elyj [ '
and a clique Q of G, setting c:= u:=e1 for all i e Q, and assigning different
vectors of B\{e1} to the remaining nodes i e V\Q, one observes that every
clique constraint is a special case of this class of infinitely many inequali-
ties. The set
TH(G) := {x e Rv I x satisfies all
orthonormal representation constraints}
is thus a convex set with
STAB(G) c TH(G) c QSTAB(G).
It turns out (see [14]) that a graph G is perfect if and only if any of the
following conditions is satisfied:
(8) TH(G)= STAB(G).

(9) TH(G)= QSTAB(G).
(10) TH(G) is a polytope.
The last result is particularly remarkable. It states that a graph is perfect if
and only if a certain convex set is a polytope.
If c E Rv is a vector of node weights, the optimization problem (with
infinitely many linear constraints)
max c x, x e TH(G)
can be solved in polynomial time for any graph G. This implies, by (8),
that the weighted stable set problem for perfect graphs can be solved in
polynomial time, and by LP duality, that the weighted clique covering
problem, and by complementation, that the weighted clique and coloring
problem can be solved in polynomial time. These results rest on the fact
that the value
S(G, c) := max {cT xx E TH(G)}
can be characterized in many equivalent ways, e.g., as the optimum value
of a semidefinite program, the largest eigenvalue of a certain set of sym-
metric matrices, or the maximum value of some function involving
orthornormal representations.
Details of this theory can be found, e.g., in Chapter 9 of [14]. The
algorithmic results involve the ellipsoid method. It would be nice to have
"more combinatorial" algorithms that solve the four optimization prob-

lems for perfect graphs in polynomial time.

Let us now move into information theory. Given a graph G = (V,E), we
call a vectorp e R a probability distribution on Vif its components sum
to 1. Let "' = (V, E")) denote the so-called n-th conormalpower of G,
i.e., V is the set of all n-vectors x = (x1,..., x,) with components x e V,
and
E":= {xy I x, y E V and 3 i with x y, E E
Each probability distribution p on Vinduces a probability distribution p"
on V as follows: p" (x) : = p (x) p(x) ..... p(x). For any node set Uc
V, let G"'[U] denote the subgraph of G") induced by Uand X (G" [U])
its chromatic number. Then one can show that, for every
0 < s < 1, the limit

HI(G, p):= li m- in log X(G( [U])
n--n p"(U), I-

exists and is independent ofs (the logs are taken to base 2). H(G,p) is
called the graph entropy of the graph G with respect to the probability
distribution p. If G = (VE) is the complete graph, we get the well-known
Shannon entropy

H(p)= plog p.
-V
Let us call a graph G = (VE) strongly splitting if for every probability dis-
tributionp on V
H(p)= H(G,p)+H(G,p)
holds. Csiszar et. al [9] have shown that a graph is perfect if and only if
G is strongly splitting.
I.e., G is perfect iff, for every probability distribution, the entropies of
G and of its complement G add up to the entropy of the complete graph
(the Shannon entropy). I recommend [9] for the study of graph entropy
and related topics.
Given all these beautiful characterizations of perfect graphs and polyno-
mial time algorithms for many otherwise hard combinatorial optimization
problems, it is really astonishing that nobody knows to date whether per-
fectness of a graph can be recognized in polynomial time. There are many
ways to prove that, deciding whether a graph is not perfect, is in NVP. But
that's all we know!
Many researchers hope that a proof of the most famous open problem
in perfect graph theory, the strongperfect graph conjecture:
A graph G is perfect ifand only if G neither contains an odd hole
nor an odd antihole as an induced subgraph.
results in structural insights that lead to a polynomial time algorithm for
recognizing perfect graphs. It is trivial that every odd hole (= chordless
cycle of length at least five) and every odd antihole (= complement of an
odd hole) are not perfect. Whenever Claude Berge encountered an imper-
fect graph G he discovered that G contains an odd hole or an odd antihole
and, thus, came to the strong perfect graph conjecture. In his honor, it is
customary to call graphs without odd holes and odd antiholes Berge
graphs. Hence, the strong perfect graph conjecture essentially reads: every
Berge graph is perfect.
This conjecture stimulated a lot of research resulting in fascinating
insights into the structure of graphs that are in some sense nearly perfect
or imperfect. E.g., Padberg [20], [21] (introducing perfect matrices and
using proof techniques from linear algebra) showed that, for an imperfect
graph G = (VE) with the property that the deletion of any node results in
a perfect graph, satisfies the following:

|V= a (G). o (G)+ 1,
G has exactly I q maximum cliques, and every node is contained in
exactly co (G) such cliques.
G has exactly I q maximum stable sets, and every node is contained
in exactly a (G) such stable sets.
QSTAB(G) has exactly one fractional vertex, namely the point
x, = 1/o (G) V v e V, which is contained in exactly | facets and
adjacent to exactly VI vertices, the incidence vectors of the maximum
stable sets.
Similar investigations (but not resulting in such strong structural
results) have recently been made by Annegret Wagler [24] on graphs
which are perfect and have the property that deletion (or addition) of any
edge results in an imperfect graph. The graph of Figure 1 is from Wagler's
Ph.D. thesis. It is the smallest perfect graph G such that whenever any
edge is added to G or any edge is deleted from G the resulting graph is
imperfect.
Particular efforts have been made to characterize perfect graphs "con-
structively" in the following sense. One first establishes that a certain class
C of graphs is perfect and considers, in addition, a finite list C of special
perfect graphs. Then one defines a set of "operations" (e.g., replacing a
node by a stable set or a perfect graph) and "compositions" (e.g., take two
graphs G and H and two nodes u e V(G) and v e V(H), define V(G o H)
: = (V(G) u V()) \{u,v} and E (Go H): = E (G u) E(G- v) u {x,y
xu e E (G), yv e E (H)} and shows that every perfect graph can be con-
structed from the basic classes C and C by a sequence of operations and
compositions. Despite ingenious constructions (that were very helpful in
proving some of the results mentioned above) and lots of efforts, this route
of research has not led to success yet. A paper describing many composi-
tions that construct perfect graphs from perfect graphs is, e.g., [8].
Chvital [7] initiated research into another "secondary structure" related
to perfect graphs in order to come up with a (polynomial time recogniza-
ble) certificate of perfection. For a given graph G = (VE), its P4-structure is
the 4-uniform hypergraph on Vwhose hyperedges are all the 4-element
node sets of Vthat induce a P4 (path on four nodes) of G. Chvital
observed that any graph whose P4-structure is that of an odd hole is an
odd hole or its complement and, thus, conjectured that perfection of a
graph depends solely on its P4-structure. Reed [23] solved Chvital's semi-
strong perfect graph conjecture by showing that a graph G is perfect iff
(12) G has the P4-structure of a perfect graph.
There are other such concepts, e.g., the partner-structure, that have
resulted in further characterizations of perfect graphs through secondary
structures. We recommend [15] for a thorough investigation of this topic.
But the polynomial-time-recognition problem for perfect graphs is still
open.
A relatively recent line of research in the area of structural perfect graph
theory is the use of the probability theory. I would like to mention just
one nice result of Promel und Steger [22]. Let us denote the number of
perfect graphs on n nodes by Perfect (n) and the number of Berge graphs
on n nodes by Berge (n), then
Perfect (n)
lim -- = 1.
-n Berge (n)

In other words, almost all Berge graphs are perfect, which means that if
there are counterexamples to the strong perfect graph conjecture, they are
very rare.

-- -' ----- -1

The theory of random graphs provides deep insights into the proba-
bilistic behavior of graph parameters (see [4], for instance). To take a sim-
ple example, consider a random graph G = (VE) on n nodes where each
edge is chosen with probality /2. It is well known that the expected values
of a(G) and (o(G) are of order log n while X(G) and X(G) both have
expected values of order nl log n. This implies that such random graphs
are almost surely not perfect. An interesting question is to see whether the
"LP-relaxation of cu(G)," the so-called fractional stability number c* (G) =
max {1Tx | x e QSTAB(G)}, is a good approximation of a(G). Observing
that the point x = (x), with x : = 1 /o(G), v e V, satisfies all clique
constraints and is thus in QSTAB(G) and knowing that o (G) is of order
log n one can deduce that the expected value of cc*(G) is of order n /log n,
i.e., it is much closer toX(G) than to c(G). Hence, somewhat surprising-
ly, c*(G) is a pretty bad approximation of c(G) in general not so for
perfect graphs, though.
To summarize this quick tour through perfect graph theory (omitting
quite a number of the other interesting developments and important

results), here is my favorite theorem:
Theorem Let G be a graph. The
and characterize G as a perfect graph:

(1) Bo (G) = X (G') for all

(2) a (G') = X (G) for all

(3) G is ape

conditions are equivalent

node-induced subgraphs G' c G.

node-induced subgraphs G' c G.

rfect graph.

(4) (o (G) a (G') I V(G')I
for all node-induced subgraphs G c G.

(5) The value max {cX x e QSTAB(G)} is integralfor allc e {0,1} .

(6) The value max {cTx x e QSTAB(G)} is integral for all c L 1.

(7) STAB(G) = QSTAB(G).

(8) TH(G)= STAB(G).

(9) TH (G)= QSTAB(G).

(10) TH(G) is apolytope.

(11) G is strongly splitting.

(12) G has the P4-structure ofa perfect graph.

References

[1] C. Berge, The history of the
perfect graphs, Southeast Asian
Bull. Math. 20, No.1 (1996)
5-10.
[2] C. Berge, Motivations and
history of some of my conjec-
tures, Discrete Mathematics
165-166 (1997) 61-70.
[3] C. Berge, V. Chvital (eds.),
Topics on perfect graphs,
Annals ofDiscrete Mathematics
21, North-Holland,
Amsterdam, 1984.
[4] B. Bollobas, Random Graphs,
[5] A. Brandstadt, V. B. Le, J. P.
Survey, SIAM, 1999.
[6] V. Chvital, On certain poly-
topes associated with graphs,
Journal of Combinatorial
Theory B 18 (1975) 138-154.
[7] V. Chvatal, A semi-strong per-
fect graph conjecture, Annals
ofDiscrete Mathematics 21
(1984) 279-280.
[8] G. Cornudjols, W H.
Cunningham, Compositions
for perfect graphs, Discrete
Mathematics 55 (1985)
245-254.
[9] I. Csiszar, J. K6rner, L.
Lovasz, K. Marton, G.
Simony, Entropy splitting for
anti-blocking corners and per-
fect graphs, Combinatorica 10
(1) (1990) 27-40.
[10] J. D.R. Fulkerson, The perfect
graph conjecture and pluper-
fect graph theorem Proc. 2nd
Chapel Hill Conf Combin.
Math. Appl, Univ. North
Carolina (1970), 171-175.
[11] D.R. Fulkerson, Blocking and
anti-blocking pairs of polyhe-
dra, Mathematical
Programming 1(1971)
168-194.
[12] D. R. Fulkerson, On the per-
fect graph theorem, Math.
Seminar, Univ. Wisconsin,

[13] M. C. Golumbic, Algorithmic
graph theory andperfect graphs,
1980.
[14] M. Gr6tschel, L. Lovisz, A.
Schrijver, Geometric algorithms
and combinatorial
optimization, Springer, Berlin-
Heidelberg, 1988.
[15] S. Hougardy, On the P4-struc-
ture ofperfect graphs, Shaker-
Verlag, Aachen 1996.
[16] S. Hougardy, Inclusions
Between Classes of Perfect
Graphs, Preprint, Institut fuir
Informatik, Humboldt-
Universitat zu Berlin, 1998,
see
lin.de/-hougardy/paper/class-
es.html>.
[17] L. Lovasz, Normal hyper-
graphs and the perfect graph
conjecture, Discrete
Mathematics 2 (1972)
253-267.
[18] L. Lovisz, A characterization
of perfect graphs, Journal of
Combinatorial Theory (B) 13
(1972) 95-98.
[19] L. Lovisz, On the Shannon
capacity of a graph, IEEE
Transf Inform. Theory 25
(1979) 1-7.
one matrices, Math.
Programming 6 (1974)
180-196.
gral polyhedra related to cer-
tain combinatorial optimiza-
tion problems, Lin. Alg. Appl.
15 (1976) 69-88.
[22] H.J. Promel, A. Steger, Almost
all Berge graphs are perfect,
Combin. Probab. Comput. 1
(1992) 53-79.
[23] B. Reed, A semi-strong perfect
graph theorem, Journal of
Comb. Theory, Ser. B 43
(1987) 223-240.
[24] A. Wagler, Critical edges in
perfectgraphs, Ph.D. thesis,
TU Berlin, 1999.

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Peaceably Coexisting

Armies of Queens

Robert A. Bosch

June 14, 1999

The 8-queens problem requires that eight
queens be placed on a chessboard in such a
way that no two can attack each other (i.e., lie
in the same row, column, or diagonal). See
Figure 1 for a solution. The 8-queens problem
was proposed by a chessplayer named Max
Bezzel in 1848. During the next two years, it
captured the attention of Carl Friedrich Gauss
(albeit briefly) and other prominent scholars.
In September 1850, Franz Nauck presented 92
solutions. See [2] for a detailed history.

Problems
1. Figure 2 demonstrates that two armies of
eight queens can peaceably coexist on a
chessboard (i.e., be placed on the board
in such a way that no two queens from
opposing armies can attack each other).
Formulate an integer program to find the
maximum size of two equal-sized, peace-
ably coexisting armies of queens.
2. How many optimal solutions does the
integer program have?

comments to lin.edu>. The most attractive
solutions will be presented in a
forthcoming issue of OPTIMA.

Pentomino Exclusion
Revisited
In the December 1998 issue of
OPTIMA, we presented an
integer programming formula-
tion of the n x n pentomino
exclusion problem. This prob-
lem requires that monominoes

Today, the problem continues to fascinate.
A web page maintained by Walter Kosters lists
66 recent books and articles that refer to the
8-queens problem or its generalization, the n-
queens problem [5]. One article, by L.R.
Foulds and D.G. Johnston, describes two
strategies for solving various "chessboard non-
attacking puzzles": maximal cliques and inte-
ger programming [3].
Here, we pose a variant.

(1 x 1 square) be placed on an n x n board in
such a way that there is no room left for any
pentominoes (objects formed by joining five
1 x 1 squares in an edge-to-edge fashion). The
goal is to use as few monominoes as possible.
We also challenged readers to find a better
couple of excellent submissions. One, by
Frank Plastria, contains a description of several
families of valid inequalities for our original
formulation as well as proofs that some of
these inequalities define facets [6]. The second,

ilk.l

Figure I

Figure 2

by Kurt Anstreicher, does not discuss the formu-
lation at all. Instead, it gives a simple, direct
proof that at least 24 monominoes are needed to
exclude all pentominoes from an 8 x 8 board
[1].
In the next section, we discuss Plastria's valid
inequalites. And then, in the three sections that
follow, we describe a completely different formu-
lation of the problem, present evidence that it is
vastly superior to the original formulation, and
show how its constraints can be used to con-
struct a simple proof of a result concerning
toroidal pentomino exclusion problems.

Valid Inequalities for the Original
Formulation
In [6], Plastria begins by noting that the origi-
nal formulation can be written very concisely:

minimize

subject to xs l for all P T
s P
x e(O,1) for alls S.

Here, S stands for the set of all squares of the
board, and P stands for the set of all pentomino
placements. Plastria considers a pentomino
placement to be a 5-element subset of S that
consists of the five squares that are covered
when some pentomino is placed either in its
"standard" orientation or in some rotated or
reflected orientation somewhere on the
board.
Plastria goes on to construct several families of
"hexomino" constraints. He proves that his hex-
omino constraints are valid inequalities and that
some are facet-generating. Figure 3 contains
graphical displays of three families of hexomino
constraints. All three are facet-generating. The
one on the left states that at least two of the six
squares in any 2 x 3 rectangular-shaped region
should receive monominoes. The center one
states that if the central square of a region that
forms a "ribbon-shaped" hexomino does not
receive a monomino, then at least two of the
other five squares of the region must receive
monominoes.

S121 2 2 1 2 2

Figure 3

Figure 4

23 3
6

1 2 3 4 5 6 7 8

Figure 5

A Completely Different For
Figure 4 displays an optimal
solution to the 8 x 8 pen-
tomino exclusion problem.
Note that in the solution,
empty squares (squares with-
out monominoes) appear in
"clumps." In fact, each clump
of empty squares is a poly-
omino ("skew" tetrominoes,
T tetrominoes, and L tetro-
minoes).

empty squares as well as portions of neighboring
squares (and portions of monominoes).
The lower lefthand corner of Figure 5 displays
the modified board. We created it by expanding
the original board (by adding a border of partial
squares) and then triangulatingg" the expansion
(by dividing each square and partial square into
equal-sized triangular pieces).
The rest of Figure 5 displays each tile in its
"standard orientation." Note that there is a tile
for each possible clump of empty squares. Each
of tiles 1 and 5 has only one orientation (hence-
forth referred to as "number 1"). Each of tiles 2,
3, and 6 has two: its standard orientation (num-
ber 1), and the orientation obtained by rotating
the standard orientation 90 degrees clockwise
(number 2). Similarly, each of tiles 4 and 9 has
7 four orientations: its standard orientation (num-
ber 1), and the orientations obtained by rotating
the standard orientation 90, 180, and 270
degrees clockwise (numbers 2, 3, and 4). Tile 7
S has eight orientations, and tile 8 has four. In
each of these last two cases, half of the orienta-
tions are reflections (orientations that result from
"picking up" another orientation and "flipping it
over").
Figure 6 demonstrates how some of these tiles
can be assembled to form the solution presented
in [4].
We now present the details of the set packing
formulation. We begin by defining a set
S= {(t,o,i,J) : tile t lies entirely on the
board if placed there in orientation o with
mulation is reference point (the small circle) in the
center of square i,j}

77 1 __

This simple fact can serve
as the foundation of a set packing formulation
of polyomino exclusion problems. (The original
formulation is a set covering formulation.) Here,
we form solutions not by placing individual
monominoes on individual squares of the board,
as in the original, set covering approach, but by
placing "tiles" on a modified version of the
board. Each tile consists of an entire clump of

Figure 6

and, for each triangular piece T of the board, a
set
C= {(t,o,i,j)eA : tile covers triangular
piece T if placed on the board in orientation
o with is reference point in the center of
square i,j}

Note that the set A tells us how and where we
can place the tiles on the board. We refer to the
elements of A as "admissible placements." (Note
that (4,4,1,1) is an admissible placement, but
(4,1,1,1), (4,2,1,1), and (4,3,1,1) are not.) And
note that the set C contains all admissible place-
ments that cover triangular piece T. We call C
the covering set for T.
The set packing formulation has a binary vari-
able for each admissible placement. In particular,
for each (t,o,i,j)e.y,

S1 if admissible placement (t, o,i,j) is used,
Y [0 otherwise.

And the set packing formulation contains one
constraint for each triangular piece T of the
board:

(t, o,) c C

These constraints make sure that each trian-
gular piece of the board is covered by at most
one tile.
We minimize the number of monominoes
placed on the board by maximizing the total
number of empty squares on the board. If, for
each tile t, vt denotes the number of empty
squares contained within (note that v, = 1, v2
2, v = v4= 3, and v5 = v6= *** = V9 = 4) then
our objective is to maximize

Figure 7

Noll
MI .... I
a INki lk IN--- Ok ii

i .... JN ..... lM _JN M ..... in ..... :111111111111L ... o n

mulation is far superior to the original, set cover-
ing formulation. The set packing formulation mino tiles (tile 9 in Figure 5).
was able to solve many more of the problems We conclude this section with a theorem
tested, and it required considerably fewer about toroidal pentomino exclusion problems.
branch-and-bound nodes on all problems tested, Theorem: For all n, all solutions to the n x n
considerably fewer iterations of the simplex toroidal pentomino exclusion problem have den-
method on all but one of the problems tested sity greater than or equal to 3/7.
(the smallest, the 6 x 6 problem), and much less Proof: First, add up all constraints of the set
CPU time on all but one of the problems tested packing formulation of the n x n toroidal prob-
(again, the smallest). lem. Since the n x n toroidal board was divided
into 8n2 triangular pieces, the resulting inequali-

Toroidal Problems
Figures 7 and 8 display two optimal solutions to
the 14 x 14 toroidal pentomino exclusion prob-
lem. (An n x n toroidal board is constructed by
gluing together both pairs of opposite borders of

a standard n x n board.) Note that each solution
Comparing the Formulations .
has a density of 3/7. (We consider the density of
We tested the two formulations on a number of a solution to be the fraction of squares of the
n x n pentomino exclusion problems, using a board that have monominoes in them.) Also
200 MHz Pentium PC and CPLEX (version note that the Figure 7 solution is made entirely
4.0.9, with all parameters at default settings). of skew tetromino tiles (tile 8 in Figure 5), while
Table 1 makes it clear that the set packing for- the Figure 8 solution is made entirely of T tetro-

Board nodus iteraticms EMOCC MdS no& s iterations sanHmds
6 x 6 1,310 1.80 5 1,217 1.30
7x 7 4,232 50,285 12.266 0 1008 120
8x9 5,106 139,745 451.34 15 4,608 12.05
9x9 ran lox 10 r rn c It 4inLmcxv 19 10.237 42.25
I Ix 11 ranimutofmesmnry 32 21,935 152.52
12 x 12 rxn Clutfin4 ruiCY 55 26,895 27431
13 x 13 ran txt Ofm mrry 402 194,724 2,976.30

(The first summation is over all triangular pieces
T.) Now let 6t denote the number of triangular
pieces in tile t. Note that 61 = 16, 82 = 32, 63 =
48, 6 = 44,6 5 = 56, 66 = 64, 6, = 60, and 6, = 69
= 56. Furthermore, note that

(Both sides count the number of triangular
pieces of the board that are covered with tiles.
The left side forms the sum by examining each
triangular piece and adding a one to the total if
the piece is covered by a tile, and a zero to the
total if it isn't. The right side forms the sum by
considering each tile used in the solution recall
that each admissible placement corresponds to
the placement of a certain tile in a certain orien-
tation in a certain position on the board and
adding the number of triangular pieces it covers
to the total.) Substituting (4) into (3) and then
dividing both sides of the resulting inequality by
14 yields.

Table 1

-- -' ----- -1

Figure 8

* []

E W

T (t,o,,J} C

I I
, (t.DolJCC

q 6

Set COwering

Set Packing

6 ,
(t,o,i,j) cA .

8n2 4n2
14 7

And since [6/141 > v for each tile t, it fol-
lows that

(t. Wo,,)A

Hence the density is greater than or equal to
3/7. O

The Pentomino Spanning Problem
In the December 1998 issue of OPTIMA, we
also challenged readers to devise an integer pro-
gramming formulation for finding the smallest
subset of pentominoes that spans an n x n
board. (A subset of pentominoes spans a board
if its members can be placed on the board in
such a way that they exclude the remaining
pentominoes.) In this section, we present a for-
mulation due to Frank Plastria [6]. For the sake
of brevity, instead of Plastria's notation we use
notation very similar to that used in the previ-
ous three sections.
We begin by numbering the pentominoes,
numbering the orientations of each pentomino,
and selecting one square of each pentomino to
serve as that pentomino's reference square. We
then define a set of admissible placements

A = {(p,o,i,j) : pentomino p lies entirely
on the board if placed there in orientation
o with its reference square on square i,j}
and, for each square i,j of the board, a covering
set
C= {(p,o,i',j') e A : pentomino p covers
square i,j, if place on the board in orienta-
tion o with its reference square on square
i',j'}.
(p,o,i,j) AS, we define a set
S,,,, = {(p',o',i,j') e A : (p',o',i',j') covers
at least one of the squares covered by
(p,o,i,j)}.

Using this notation, we can write Plastria's
formulation as follows:

min yp.o,i,j

s.t. Y yP.o.i < 1
(t o' ' j') C 1

I yp,o,i,j 1
(po,i, j) l)a

Y Yp,,.,s + Z
('pcf (C'.',tj') &s
(p, o',i', j) A
yop, e{0,1}

l1< ,j
1< p<12

yI, of

(p, o,i, j) e A

The variables are basically the same as in the set
packing formulation of the pentomino exclu-
sion problem: for each (p,o,i,j)e A',

f1 if admissible placement (p, o, i, j) is used,
YP'oiJ = otherwise.

The first set of constraints ensures that each
square is covered by at most one pentomino.
The second set guarantees that each pentomino
is placed on the board at most once. The third
set makes sure that the pentominoes that are
placed on the board actually span the board. In
particular, the constraint corresponding to
admissible placement (p,o,i,j) makes sure that if
pentomino p is not placed on the board, then at
least one of the squares that wouldhave been
covered by (p,o,i,j) is covered. To see this, note
that
1. the leftmost summation takes on a value of
0 if and only if pentomino p is not placed
on the board, and
2. the rightmost summation takes on a posi-
tive value if and only if at least one of the
squares that would have been covered by
(p,o,i,j) is covered.

References

[1] K.M. Anstreicher, A pentomino exclusion problem, preprint, University of Iowa, February 1999.
[2] P.J. Campbell, Gauss and the eight queens problem: a study in miniature of the propogation of
historical error, Historia Mathematica, 4 (1977), pp. 397-404.
[3] L.R. Fould and D.G. Johnston, An application of graph theory and integer programming: chess-
board non-attacking puzzles, Mathematics Magazine, 57 (1984), pp. 95-104.
[4] S.W Golomb, Polyominoes: Puzzles, Patterns, Problems, and Packings (Princeton University
Press, Princeton, NJ, 1994).
[5] W. Kosters, n-Queens web page, .
[6] F. Plastria, On IP formulations for the pentomino exclusion and spanning problems, preprint,
Vrije Universiteit Brussel, May 1999.

-- -' ----- -1

) Second International Workshop on Approximation Algorithms for Combinatorial Optimization Problems,
and Third International Workshop on Randomization and Approximation Techniques in Computer Science
August 8-11, 1999,Berkeley, CA,USA
http://cuiwww.unige.ch/-rolimlapprox;http://cuiwww.unige.chl~rolim/random
) Symposium on Operations Research 1999, SOR'99
September 1-3, 1999,Magdeburg,Germany
http://www.uni-magdeburg.de/SOR991
) Sixth International Conference on Parametric Optimization and Related Topics
October 4-8, 1999,Dubrovnik,Croatia
http://www.math.hr/dubrovniklindex.htm
) INFORMS National Meeting
)7th INFORMS Computing Society Conference on Computing and Optimization: Tools for the New Millenium
January 5-7, 2000,Cancun,Mexico
) DIMACS 7th Implementation Challenge:Semidefinite and Related Optimization Problems Workshop
January 24-26, 2000,Rutgers University; Piscataway, NJ
http://dimacs.rutgers.edu/Workshops/7thchallenge
) Seventh International Workshop on Project Management and Scheduling (PMS 2000)
April 17-19, 2000,University of Osnabrueck,Germany
http://www.mathematik.uni-osnabrueck.de/researchlOR/pms2000/
) Applied Mathematical Programming and Modelling Conference (APMOD 2000)
17-19 April 2000, Brunel University, London
http://www.apmod.org.uk
) ISMP 2000 17th International Symposium on Mathematical Programming
August 7-11,2000,Georgia Institute of Technology, Atlanta,GA,USA
http://www.isye.gatech.edu/ismp2000

ov

T- f

v f

First Announcement
General Information
Call For Papers

The 17th International Symposium on Mathematical Programming

August 7-11, 2000

Georgia Institute of Technology, Atlanta, Georgia, USA.

http://www.isye.gatech.edu/ismp2000

Papers on all theoretical, computational
and practical aspects of mathematical
programming are welcome. The presen-
tation of very recent results is encour-
aged.

submission of titles and abstracts, and

Site The symposium will take place at
the Georgia Institute of Technology
located in Atlanta, Georgia, USA. Hotels
of various categories will be available as
well as low priced accommodations on
the Georgia Tech campus.

Organizing Committee
General Co-Chairs: G.L. Nemhauser,
M. WP. Savelbergh
Program Committee Co-Chairs:
E.L.Johnson, A. Shapiro, J. VandeVate,
R. Monteiro, R. Thomas, C. Tovey,
V Vazarani;
Local Committee Chair: D. Llewellyn;
Committee Members: Al-Khayyal
E. Barnes, P. Keskinocak, A. Kleywegt,
J Sokol
Structure of the Meeting A large number
of people will be invited to organize ses-
sions. Others may nominate themselves
as session organizers. To do so, they must
write us and, upon our program com-
mittee's approval, they will be included in
the list.
During the plenary opening session, the
following prizes will be awarded:
Dantzig Prize (for original research hav-
ing a major impact on mathematical pro-
gramming); Fulkerson Prizes (for out-

standing papers in discrete mathematics);
Orchard-Hays Prize (for excellence in
computational mathematical program-
ming); and A.W. Tucker Prize (for an
outstanding paper by a student).
Morning Historical Perspectives
Plenaries Nonlinear Programming:
Roger Fletcher; Integer Programming:
Martin Gritschel; Combinatorial
Optimization: .1 Pulleyblank;
Linear Programming: Michael Todd4
Developments in the Soviet Union:
Boris Polyak.
Afternoon Semi-Plenaries Aharon
Ben-Tal, .1 Cunningham,
Don Goldfarb, Alan King, Jim Orlin,
James Renegar, Alexander Schrijver,
David \ Yinyue Ye.

Social Program
Sunday, August 6: Welcome reception,
18:00 19:30
Monday, August 7: Opening ceremony,
9:00 12:00
Wednesday, August 9: Evening reception,
18:00 23:00
A special program for accompanying
persons is being organized.

Mailing address: ISMP 2000, c/o A.
Race, School of Industrial and Systems
Engineering, Georgia Institute of
Technology, Atlanta, GA 30332-0205,
USA;
Fax: (+1) 404 894 0390;
E-Mail: ;
Web site:
.

List of Topics
Sessions on the following topics are planned.
Suggestions for further areas to be included are welcome.
(A) Approximation Algorithms,
(B) Combinatorial Optimization,
(C) Complementarity and Variational Inequalities,
(D) Computational Biology,
(E) Computational Complexity,
(F) Computational Geometry,
(G) Convex Programming and Nonsmooth Optimization,
(H) Dynamic Programming and Optimal Control,
(I) Finance and Economics,
(J) Game Theory,
(K) Global Optimization,
(L) Graphs and Networks,
(M) Integer and Mixed Integer Programming,
(N) Interior Point Algorithms and Semi-Definite Programming,
(0) Linear Programming,
(P) Logistics and Transportation,
(Q) Multicriteria Optimization,
(R) Nonlinear Programming,
(S) Parallel Computing,
(T) Production Planning and Manufacturing,
(U) Scheduling,
(V) Semi-Infinite and Infinite Dimensional Programming,
(W) Software,
(X) Statistics,
(Y) Stochastic Programming, and
(Z) Telecommunications and Network Design.

Call for Proposals to Host

I S M P 2 00 3

The time has come for all interested parties to make propos-
als for hosting the 2003 International Symposium on
Mathematical Programming. Following tradition, a university
site outside the US will host the 2003 Symposium.
All proposals are welcome and will be examined by the
Symposium Advisory Committee, composed of Karen
Aardal, John Dennis, Martin Grotschel and Thomas Liebling
(Chair). It will make its recommendation based on criteria
such as professional reputation of the local organizers, facili-
ties, accommodations, accessibility and funding. Based on the
recommendations of the Advisory Committee, the final deci-
sion will be made and announced by the MPS Council dur-
ing the 2000 Symposium in Atlanta.
Detailed proposal letters should be addressed to: Prof.
Thomas M. Liebling, DMA-EPFL, CH-1015 Lausanne,
Switzerland (E-mail: Thomas.Liebling@epfl.ch).

Workshop on Discrete
Optimization (DO'99)
DIMACS/RUTCOR, Rutgers
University

July 25-30, 1999

http://rutcor.rutgers.edu/~do99
The updated announcement below
includes, besides the information
contained in the first announce-
ment, new information concerning:
lines); 2. Support for participants
(including the availability of an
NSF grant); 3. Accommodations
lines); and 4. Publications.

Goals Discrete optimization under-
went a tumultuous development in
the last half century and had a par-
ticularly spectacular growth in the
last few decades. The main goal of
this workshop is to survey the state
of the art in discrete optimization.
This goal will be achieved by the
presentation of expository lectures
presenting the major subareas of
the field, including its theoretical
foundations, its methodology and
applications. The surveys will be
presented by some of the most
prominent researchers in the field.
DO'99 will also provide a forum
for the presentation of new devel -
opments in discrete optimization.
To accomplish this goal, DO'99
will feature a series of sessions for
the presentation of contributed
talks, presenting the latest research
of the participants.
DO'99 is being held 22 years after
goals, and whose collection of sur-
veys (Discrete Optimization I and
II, Annals of Discrete Mathematics,
vols. 4 and 5, P.L. Hammer, E.L.
Johnson and B.H. Korte, eds.,
North Holland, Amsterdam, New
York, Oxford, 1979) is still fre-
quently used. It is hoped that
DO'99 will present the latest state
of the art in discrete optimization
and will provide a similarly useful
source of information and inspira -
tion to the community of discrete
optimizers as DO'77 did 22 years
ago.

Venue The DO'99 workshop will
take place on the Busch Campus of
Rutgers, The State University of
New Jersey, July 25-30, 1999.

Invited SurveyTalks Egon Balas
(Carnegie Mellon University), Lift-
and-project: progress and some
open questions; Peter Brucker
(University of Osnabrueck),
Complex scheduling problems;
Rainer Burkard (Technical
University of Graz), Assignment
problems; Vasek Chvatal (Rutgers
University), On the solution of
traveling salesman problems;
Gerard Cornuejos (Carnegie Mellon
University), Packing and covering;
Yves Crama (University of Liege),
Optimization models in produc-
tion planning; Fred Glover
search & evolutionary methods:
unexpected developments; Alan
I.' (IBM Research Center),
Greedy algorithms in linear pro-
gramming problems; Karla
I. (George Mason
University), Applications of col-
umn-generation and constraint-
generation methods; Toshihide
Ibaraki (Kyoto University), Graph
connectivity & its augmentation;
BernhardKorte (University of
Bonn), Gigahertz-processors need
discrete optimization; Jakob Krarup
(University of Copenhagen),
Locational decisions with friendly
and obnoxious facilities; Tom
Magnanti (M.I.T.), Network
design; Silvano Martello (University
of Bologna), Bin packing problems
in two and three dimensions;
George L. Nemhauser (Georgia
Institute of Technology), Discrete
Optimization in Air
Transportation; Jim Orlin and Ravi
Ahuja (M.I.T.),Neighborhood
Pulleyblank (IBM Research
Center), Hilbert bases,
Caratheodory's theorem and integer
programming; Andras Recski
(Technical University of Budapest),
Combinatorics of grid-like frame-
works; Paolo Toth (University of
Bologna), Vehicle routing; David
.1 (IBM Research Center),
Approximation algorithms; and
Laurence Wolsey (Catholic
University of Louvain), Survey on
mixed integer programming.

Workshop on theTheory and Practice of Integer Programming
in Honor ofRalph E. Gomory
on the Occasion of his 70th Birthday

T Iare pleased to announce a workshop in celebration of Ralph
SGomory's 70th birthday. The focus of the workshop will be on
S integer linear programming. The workshop is sponsored by
DIMACS, as part of the 1998-99 Special Year on Large-Scale Discrete
Optimization, and by IBM. The workshop will be held August 2-4, 1999,
at the IBM Watson Research Center in Yorktown Heights, New York. The
workshop will include lectures by leading international experts covering all
aspects of integer programming. We hope that the lecture program will be
of particular interest to young researchers in the field, including Ph.D. stu-
dents and post-doctoral fellows.
A conference banquet will be held with Alan I.' (IBM) as the Master
of Ceremonies. The banquet speakers will include Paul Gilmore (University
of British Columbia), Ellis Johnson (Georgia Tech), and Herb Scarf(Yale).
Invited Lecturers include: Karen I. Aardal, Utrecht University; Egon Balas,
Carnegie Mellon University; Francisco Barahona, IBM Watson Research
Center; Imre I Hungarian Academy of Sciences; Daniel Bienstock,
Columbia University; Robert *,. Rice University; Charles E. Blair,
University of Illinois; Vasek Chvatal, Rutgers University; Sebastian Ceria,
Columbia University; Gerard Cornuefols, Carnegie Mellon University;
A. H. Cunningham, University of Waterloo; Johnj Forrest, IBM
Watson Research Center; MichelX. Goemans, Universite Catholique de
Louvain; Ralph Gomory, Sloan Foundation; Peter Hammer, Rutgers
University; TC. Hu, University of California at San Diego; Ellis ohnson,
Georgia Tech; MikeJuenger, Universitat zu Koeln; BerhardKorte,
University of Bonn; Thomas L. Magnanti, Massachusetts Institute of
Technology; George L. Nemhauser, Georgia Institute of Technology; Gerd
Reinelt, Universitat Heidelberg; Martin WP.Savesbergh, Georgia Institute
of Technology; Herbert E. Scarf Yale University; Andras Sebo, University of
Grenoble; Bruce Shepherd, Lucent Bell Laboratories; BerndSturmfels,
University of California at Berkeley; Mike Trick, Carnegie Mellon
University; Leslie Earl Trotter, Jr., Cornell University; Robert Weismantel,
University I. -.i-... .. DavidP .1 IBM Watson Research
Laboratory; Laurence Alexander .T I. University Catholique de Louvain;
and Giinter Ziegler, Technische Universitat Berlin.
Conference Organizers: .1 Cook, Rice University; and .1
Pulleyblank, IBM Watson Research Center.
.

-- -' ----- -1

DIMACS 7th Implementation Challenge:
Semidefinite and Related Optimization Problems Workshop
DIMACS Center, CoRE Building, Rutgers University; Piscataway, NJ
January 24-26, 2000

University; David Johnson, AT&T Labs -
Research; Gabor Pataki, Columbia University
Presented under the auspices of the Special
Year on Large Scale Discrete Optimization.
The purpose of DIMACS computational chal -
lenges has been to encourage the experimental
evaluation of algorithms, in particular those with
efficient performance from a theoretical point of
view. The past Challenges brought together
researchers to test time-proven, mature, and
novel experimental approaches on a variety of
problems in a given subject. As the subject of
the last Challenge of this century, one could
hardly think of a better choice than Semidefinite
Programming (SDP), one of the most interest-
ing and challenging areas in optimization theory
to emerge in the last decade. In the past few
years, much has been learned on both the kinds
of problem classes that SDP can tackle, and the
best SDP algorithms for the various classes. In
limits of the current approaches to solving
SDP's.
A closely related problem to semidefinite pro-
gramming is that of convex quadratically con-
problem resides in between linear and semidefi-
nite programming. It also arises in a variety of
applications from statistics to engineering; and a
number of combinatorial optimization prob-
lems, in particular in the Steiner tree problems
and plant location problems, have found QCQP
as a subproblem. Similar to, and indeed by an
extension from, semidefinite programming, a
great deal is known about optimization with
convex quadratic constraints as well as limitation
of current methods. Finally, this knowledge has
been extended to problems containing variables
and constraints with some or all of linear, con-
This Challenge attempts to distill and expand
upon this accumulated knowledge.
We have collected a variety of interesting and
challenging SDP instances in the following class-
es. We have made an effort to create a collection
containing instances that are as "real" as possi-
ble, are presently on, or beyond the limits of
solvability, and whose solution would expand
our knowledge on the applicability of SDP.
More precisely, we included: MAXCUT prob-
lems from theoretical physics currently solvable
by polyhedral, but not by semidefinite methods;
the Lovisz-Schrijver semidefinite relaxations of
0-1 MIP's, which are unsolvable by either cut-

ting plane methods, or branch and bound; truss
topology, and Steiner tree problems lacking a
strictly complementary solution.
We invite papers dealing with all computational
aspects of semidefinite programming and related
problems. In particular, the following classes of
problems are of special interest: (1) Cut, and
partition problems; (2) Theta function, and
graph entropy problems; (3) SDP relaxation of
very difficult, (currently unsolvable) 0-1 mixed
integer programming problems; (4) Problems in
grams from engineering; (5) Problems from sta-
tistics and finance; (6) SDP instances from engi-
neering, for example truss topology design, and
control theory problems; (7) Difficult, randomly
generated problems designed to challenge algo-
rithms on performance and numerical stability;
(8) In addition to the classes above, we invite
investigations which focus on using SDP and
related codes to test out behavior of heuristics
and other new applications. The SDP code can
be developed by the investigators or they may
choose off-the-shelf codes.
All communications regarding the challenge
should be directed to rutgers.edu>; in particular, preliminary proposal
submission, extended abstracts, and possible
submission of software and problem instances
should be sent to the above address.

Important Dates September 15 -Preliminary pro-
posals due for comment and feedback; November
15 Extended abstracts due date for consideration
for the workshop;January 24-26- The workshop
will take place; Final drafts due date for appear-
ance in the workshop proceedings will be deter-
mined.

Further Information There will be a \$40/day,
\$5/day workshop registration fee for postdocs
and graduate students. For information on regis-
tration, travel and accommodations, please visit
the workshop web site
7thchallenge>; the conference e-mail address is
.

APMOD 2000
The Applied Mathematical Programming
and Modelling Conference (APMOD 2000)
Brunel University, London,
17-19 April 2000
The objective of APMOD 2000 is to bring
together active researchers, research students
and practitioners from various countries and
provide a forum for discussing and presenting
established as well as new techniques in
Operational I .... I. i 1 i. !i-.. i.i.. iir Science.
APMOD 2000 provides a bridge between
emerging research and their applicability in
industry. The theme of APMOD 2000 is
chosen to be Corporate Application of
Mathematical Optimisation. The conference
embraces a broad range of classical OR topics
including linear programming, integer pro-
gramming, combinatorial optimisation, sto-
chastic programming and nonlinear program-
ming. Emphasis is placed on novel techniques
of approaching these problems and their
application in industries such as finance, the
utilities, transport and many other diverse
areas of interest.
The three-day event will have plenary sessions
by leading researchers and also four parallel
streams. The written contributions will be
refereed and published in the Annals of
Operations Research.
For further information and pre-registration,
contact ; alternatively,
write to: Mrs Gail Woodley, APMOD 2000,
Department of Mathematics and Statistics
Brunel University, Uxbridge, Middlesex, UB8
3PH, United Kingdom.

Symposium on Operations Research
1999, SOR '99
During September 1-3, 1999, an
International Symposium, SOR'99, organ-
ized by the German Operations Research
Society (GOR) will take place in Magdeburg,
Germany. All areas of Operations Research
will be covered at this conference. For more
information, contact: G. Schwodiauer (gener-
al chair), University .- I ,i. .-i.-., Faculty
of Economics and Management, P.O. Box 41
20, D-39016 Magdeburg, Germany; phone
+49 391 6718739; fax +49 391 6711136;
about the conference can be found online at

.

I

First Announcement and Call for Papers

Seventh International Workshop on Project Management and
Scheduling (PMS 2000)
April 17-19, 2000
University of Osnabrueck, Germany

Following the six successful work-
shops in Lisbon (Portugal), Como
(Italy), Compiegne (France),
Leuven (Belgium), Poznan
(Poland), and in Istanbul (Turkey),
the Seventh International
Workshop on Project Management
and Scheduling is to be held in
Osnabriick, a small, charming city
located halfway between Cologne
and Hamburg.
The main objectives of PMS 2000
are to bring together researchers in
the area of project management
and scheduling in order to provide
a medium for discussions of
research results and research ideas
and to create an opportunity for
researchers and practitioners to get
involved in joint research.
Another objective is to attract new
recruits to the field of project man-
agement and scheduling to make
them feel a part of a larger net-
work. For this aim there will be
special sessions on railway schedul-
ing, timetabling, batch scheduling
in the chemical industry, and robot
scheduling.
Program Committee Peter Brucker,
Chair (University of Osnabrueck),
Lucio Bianco (IASI, Rome), Jacek
Blazewicz (Poznan University of
Technology), Fayez Boctor (Laval
University), Jacques Carlier
(Universiti de Technologie
Compiegne), Eric Demeulemeester
(Katholieke Universiteit Leuven),
Andreas Drexl (Christian-Albrechts-
Universitat zu Kiel), Salak E.
1 (North Carolina State
University), Selcuk Erenguc
(University of Florida), Willy
Herroelen (Katholieke Universiteit
Leuven), Wieslaw Kubiak
(Memorial University of
Newfoundland), C ee Lee
(Texas A&M University), Klaus
Neumann (University of
Karlsruhe), Linet Ozdamar
(Istanbul Kultur University),James
Patterson (Indiana University),
Erwin Pesch (University of Bonn),
Marie-Claude Portmann (Ecole des
Mines de Nancy, INPL), Avraham
Shtub (Technion Israel Institute of
Technology), Roman Slowinski

(Poznan University ofTechnology),
Superior Technico, Lisbon),
Gunduz Ulusoy (Bogazici
University, Istanbul), Vicente Vails
(University of Valencia), Jan
A (Poznan University of
Technology), and RobertJ .1
(Monash University).

Preregistration If you are interest-
web site
rueck.de/research/0R/pms2000/>
and complete the pre-registration
uni-osnabrueck.de>.
Pre-registration does not involve
any obligations, but helps us to
plan the schedule and keep you
name(s), affiliation and e-mail
address, and whether or not you
intend to give a talk. Presentations
will be selected on the basis of a
three-page extended abstract to be
submitted no later than September
15, 1999.
Important Dates Abstract submis-
sion: September 15, 1999;
November 1, 1999; Workshop reg-
1999.
Registration Costs include the
conference fee, a welcoming party,
coffee breaks, and three lunches.
The following prices are provision-
al: Early registration fee, DM 300;
Late registration fee, DM 350;
Excursion and dinner, to be
announced.
is December 15, 1999. Please con-
sult the conference web site to reg-
ister.
Information Sources For up-to-
date information, including infor-
mation on hotels and the city of
site
rueck.de/research/OR/pms2000/>

istered jointly by the Mathematical Programming Society (MPS)
and the Society for Industrial and Applied Mathematics (SIAM).
This prize is awarded to one or more individuals for original
research which by its originality, breadth and depth, is having a
major impact on the field of mathematical programming. The contribu-
tion(s) for which the award is made must be publicly available and may
belong to any aspect of mathematical programming in its broadest sense.
Strong preference will be given to candidates that have not reached their
50th birthday in the year of the award.
The prize will be presented at the Mathematical Programming Society's
triennial symposium, to be held 7-11 August 2000, in Atlanta, Georgia,
USA. Past prize recipients are listed on the MPS web site
. The members of the prize com-
mittee are William H. Cunningham, Claude Lemarechal, Stephen M.
Robinson (Chair), and Laurence A. Wolsey.
Nominations should consist of a letter describing the nominee's qualifi-
cations for the prize, and a current curriculum vitae of the nominee
including a list of publications. They should be sent to: Stephen M.
Robinson, Department of Industrial Engineering, University of
USA, E-mail: .
Nominations must be received by 15 October 1999. Any nominations
received after that date will not be considered. Submission of nomination
materials in electronic form (e-mail with attachments as needed) is strong-
ly encouraged.

Linear Semi-Infinite Optimization

Miguel A. Goberna and Marco A. Lopez

John Wiley & Sons, 1998
ISBN 0-471-97040-9

L inear semi-infinite programming deals with the problem of minimiz-
ing (maximizing) a linear objective function of a finite number of
variables with respect to an (possibly and generally) infinite num-
ber of linear constraints. There is a great variety of applications of
semi-infinite optimization, including problems in approximation
theory (using polyhedral norms), operation research, optimal control,
boundary value problems and others. These applications and appealing
theoretical properties of semi-infinite problems gave rise to intensive (and
up to now undiminished) research activities in this field since their incep-
tion in the 1960s.
The book under review is, according to the authors, intended "... as a
monograph as well as a textbook..." It fills a gap in the present literature
The authors organize their material into four parts:
Part I, Modeling, deals with numerous examples of occurring linear
semi-infinite optimization problems, divided into two chapters (1)
Modeling with the primal problem and (2) Modeling with the dual
problem. Most of the models described in Part I arise from other fields of
applied mathematics.
In Part II, Linear Semi-Infinite Systems, the feasible sets of (primal)
linear semi-infinite optimization problems are investigated. It provides the
necessary fundamentals for the forthcoming theory and contains the
chapters (3) Alternative Systems, (4) Consistency, (5) Geometry and (6)
Stability.

Graph Theory

Reinhard Diestel

Springer Verlag
ISBN 0-387-98210-8

Have you heard of Szemeridi's regularity lemma? If you have been in
touch with graph theory at least a little, the answer is almost sure-
ly yes. How about its proof? In case you were so far too afraid to
go through the details and understand the basic ideas of this far-
reaching result, the book Graph Theory by G. Diestel will prove an
excellent guide. One of the main strengths of this book is making deep and
difficult results of graph theory accessible.
Actually, the goal of this book is (at least) twofold: First, to give a solid
introduction to basic notions and provide the standard material of graph
theory, such as the matching theorems of Konig, Hall, Tutte, Petersen, the
theorems of Dilworth and Menger on paths and chains, Kuratowski's char-
acterization of planar graphs, coloring theorems of Brooks and Vizing, etc.
But already the introductory chapters include results which have not yet
appeared in textbooks: Mader's theorem (1.4.2) on the existence of k-con-
nected subgraphs of a sufficiently dense graph, Tutte's characterization of 3-
connected graphs. It is also refreshing to see the proof of Mader's theorem
(3.6.1) on the existence of large topological complete graph as a minor in
a dense graph, or of the pretty theorem of Jung and of Larman and Mani
(3.6.2) stating the existence of k disjoint paths between any set of k pairs of
nodes in a sufficiently highly connected graph.
The main novelty of this book, however, is that a great number of diffi-
cult and deep theorems, along with their full proofs, are exhibited. These
include some older results like Szemeridi's above-mentioned regularity
lemma, or Fleischner's theorem (10.3.1) on the Hamiltonicity of the square
of a graph, with a full proof of over seven pages. A fundamental result

The essential Part III, Theory of Linear Semi-Infinite Programming,
presents in chapter (7) Optimality a natural extension of the classical
Karush-Kuhn-Tucker theory to linear semi-infinite optimization problems
and provides further information on some optimality conditions for the
dual problem and for convex semi-infinite programming problems.
Chapter (8), Duality, develops a systematic approach to the main topics
of duality theory in linear semi-infinite programming. Additionally, the
(occasionally) occurring duality gaps are analyzed, as well as the connection
between duality gaps and the applicability of discretization methods.
Chapter (9), Extremality and Boundedness, deals with extreme points
and extreme directions of the feasible sets and optimal sets of a (fixed) pair
of linear semi-infinite optimization problems.
Perturbations in the data set and their impact on the solutions are inves-
tigated in chapter (10), Stability and Well-Posedness, with an analysis of
the optimal value function and the optimal-set mapping.
Part IV, Methods of Linear Semi-Infinite Programming, gives in the
two chapters (11) Local Reduction and Discretization Methods and (12)
Simplex-Like and Exchange Methods an overview of numerical methods
for solving linear semi-infinite optimization problems. "They are described
in a conceptual form... but omitting a detailed discussion of the numerical
difficulties encountered in the auxiliary problems."
Each chapter contains a lot of historical and bibliographical notes and
hints, and ends with a collection of exercises (without solutions). These
exercises include routine tasks and applications as well as theoretical com-
plements.
For convenience, some basic concepts and properties of convex sets and
convex functions are collected in an Appendix (without proofs). So the text
is nearly self-contained.
The book gives a good view of the topic. It is addressed to (graduate) stu-
dents in mathematics and to scientists who are interested in the ideas
behind the theory of linear optimization. The reader is assumed to be
familiar with linear algebra and elementary calculus; he should have a cer-
tain knowledge in linear programming and also in elementary topology.
The text is carefully written, the exposition is clear and goes quite deeply
into details. The book is more to provide a profound discussion of the sub-
ject than to get a first insight into the topic. It may be also used as a basis
and a guideline for lectures on this subject; the authors give some propos-
als of how to arrange the material for several courses.
As a minor complaint it should be mentioned that the list of Symbols
and Abbreviations (p. 321) unfortunately does not contain the place (page)
where certain notation occurs or is defined, so the reader sometimes has
trouble finding it.
All in all, the book leaves a remarkable impression of the concepts, tools
and techniques in linear semi-infinite optimization. Students as well as pro-
fessionals will profitably read and use it.
FRIEDRICH JUHNKE, MAGDEBURG

(Theorem 9.3.1) (due to three groups of authors around the beginning of
the '70s) from induced Ramsey theory is also completely proved (four
pages). The classical results on algebraic flows, including Tutte's investiga-
tions and Seymour's 6-flow theorem, are fully covered as well. Not only
Lovisz's perfect graph theorem (5.5.3), stating the perfectness of the com-
plement of a perfect graph, is discussed, but his second, significantly deep-
er characterization of perfect graphs, as well (Theorem 5.5.5). (Naturally,
not every fundamental result of graph theory could be included with its
proof; but perhaps if the reviewer may make a suggestion the author
may find a way in the next edition to exhibit a proof of Mader's beautiful
theorem (3.4.1) on the maximum number of independent H-paths.)
Beyond these classics, recent or even brand-new results are also discussed.
For example, Galvin's charming list-colouring theorem (5.4.4) has appeared
in 1995, or Thomassen's pearl on 5-choosability of planar graphs in 1994.
An even more recent, very difficult result (Theoerem 8.1.1), due to Koml6s
and Szemeridi and to Bollobas and Thomason, state that every graph with
average degree at least cr contains K' as a topological minor. The paper of
Koml6s and Szemeridi appeared in 1996 while the work of Bollobas and
Szemeridi has appeared only this year. Isn't it nice that a proof (of more
than five pages) is already accessible in Diestel's book?
A grand undertaking of the past 15 years has been the proof of the so-
called minor theorem (what a contradiction between name and signifi-
cance!) It reads: in every infinite set offinite graphs there are two such that one
is the minor of the other. The supporting theory of tree-decompositions,
well-quasi-orderings, minors has mainly been developed by N. Robertson
and P. Seymour. The complete proof of the minor theorem is over 500
pages, so not surprisingly this is excluded here, but Chapter 12 on Minors,
Trees and WQO offers an excellent overview of the framework of the proof
along with its far-reaching consequences.
The book includes 12 chapters: (1) The Basics, (2) Matching, (3)
Connectivity, (4) Planar Graphs, (5) Colouring, (6) Flows, (7)
Substructures in Dense Graphs, (8) Substructures in Sparse Graphs, (9)
Ramsey Theory for Graphs, (10) Hamiltonian Cycles, (11) Random
Graphs, and (12) Minors, Trees, and WQU.
As far as the basic approach of the book is concerned, I just cannot state
it any better than the back-page review of the book does: "Viewed as a
branch of pure mathematics, the theory of finite graphs is developed as a
coherent subject in its own right, with its own unifying questions and
methods. The book thus seeks to complement, not replace, the existing
more algorithmic treatments of the subject."
The book is written in a thorough and clear style. The author puts spe-
cial emphasis on explaining the underlying ideas, and the technical details
are made as painless as possible. Some typographic novelties are introduced:
for example, there are little reminders on the margins to notations, defini-
tions, references. These may be helpful in following the text more easily; the
general outlook of a page, however, becomes sometimes a bit messy.
All in all, the book of R. Diestel is a smooth introduction to standard
material and is a particularly rich source of deep results of graph theory. I
can highly recommend it to graduate students as well as professional math-
ematicians.
ANDRAS FRANK

-- -' ----- -1

New Optimization Book Series

J.E. Dennis, Jr.

Our MPS Council and the SIAM
Council and Board of Trustees have
approved a new joint book series, the
MPS/SIAM Series on Optimization.
Both professional societies will be
responsible jointly for the scientific content
of the series for which SIAM will act as pub-
lisher. The series was announced at the May
the first manuscripts submitted to the series
are in review. Our first volume should appear
early in 2000. SIAM has agreed that MPS
members will receive the SIAM member dis-
count on all SIAM books, not just the books
in this series.
We are in the process of naming the edito-
rial board, which will be organized in the
standard MPS format. The first appointment
other than my own as Editor-in-Chief, was
the appointment of Steve Wright as Co-
Editor for Continuous Optimization. I con-
sidered Steve's agreement to undertake this
new task a key issue for the success of the
series. He is responsible, hard-working, and
he has written a fine monograph on interior
point methods. Thus, he understands the
huge difference between writing a paper and
writing a successful book. We are in the
process of naming a Co-Editor for Discrete
Optimization, and then we will make collec-
tive decisions concerning the Associate
Editors. I call the new series MPC, and I
think of it as the logical next step after MPA
and MPB. I hope you will agree, and help to
make the series a success by sending us your
manuscripts and acting as a reviewer on
request.
Our aim will be to publish three to five
high quality books each year. The series will
cover the entire spectrum of optimization.
We welcome research monographs, textbooks
at all levels, books on applications, and tuto-
rials. Because our goal is to complement
MPA and MPB, we do not intend to publish
proceedings or collections of papers.
The requirements for the series are simple:
a manuscript must advance the understand-
ing and practice of optimization, and it must
be clearly written appropriately to its level.
The content must be of high scientific quali-
ty. Our review process will be somewhere

between that of a journal and a commercial
book publisher. The journal review process
has a certain "gatekeeper" aspect to it, appro-
priate because of the significance that aca-
demic committees attach to journal publica-
tion records. Our referee process will also be
judgmental, but we intend to be more con-
sultative with authors than is common for
journal editors. We will encourage and work
with the authors if we feel that the manu-
script belongs in the series. Because of our
high standards, we hope that publication in
our series will mean more to an author's vita
than books generally mean.
The advantages of SIAM as a scientific
partner are self-evident. However, I feel
strongly that the case for SIAM as publisher
is as compelling. SIAM publishes a wide vari-
ety of books, and the quality of books they
publish has been always at least as high as the
quality of commercial publishers, and now it
seems generally higher. SIAM markets aggres-
sively at meetings, online, and by mail. At
the same time, books published with SIAM
stay in print long after a commercial publish-
er would have allowed them to go out of
print. The prices are consistently lower than
for commercial publishers, though the royal-
ties are competitive. SIAM will maintain a
web site for misprints and comments on each
book in the series. I believe that this service
will be especially valuable for textbooks
because instructors can share useful software
Of course, it would have been possible to
collaborate with a commercial publisher in
this venture; however, frankly, I doubt the
future of commercial publishers for scientific
books. At the Atlanta meeting next year,
compare the per page cost at the various
booths and you will wonder, as I do, if com-
mercial scientific publishers have a future
except for elementary texts. Let me be clear
that we hope to sell a lot of books, but we
plan to publish fine books at low prices and
to keep them in print. The point here is that
a commercial publisher is accountable to
investors, while SIAM is accountable to its
members, people like us.

For the electronic version of OPTIMA, please see:

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FIRST CLASS MAIL

EDITOR:
Karen Aardal
Department of Computer Science
Utrecht University
PO Box 80089
3508 TB Utrecht
The Netherlands
e-mail: aardal@cs.ruu.nl
URL: http://www.cs.ruu.nl/staff/aardal.html

AREA EDITOR, DISCRETE OPTIMIZATION:
Sebastian Ceria
417 Uris Hall
Columbia University
New York, NY 10027-7004
USA
e-mail: sebas@cumparsita.gsb.columbia.edu
URL: http://www.columbia.edu/~sc244/

BOOK REVIEW EDITOR:
Robert Weismantel
Universitat Magdeburg
Fakultat fir Mathematik
Universitatsplatz 2
D-39106 Magdeburg
Germany
e-mail: weismant@math.uni-magdeburg.de

Donald W. Hearn, FOUNDING EDITOR
Elsa Drake, DESIGNER