O P T I M A Mathematical Programming Society Newsletter JUNE19 9 9
Ch a r a c t e r i z a t i o n s of Per fect Gra p h s by Martin Grötschel
62 O P T I M A 6 2 JUNE 19 9 9 2 Ch a r a c t e r i z a t i o n s of Per fect Gra p h s he favorite topics and results of a researcher change over time, of course. One area that I have always kept an eye on is By Martin Grötschel that of perfect graphs. These graphs, introduced in the late ‘50s and early ‘60s by Claude Berge, link various mathemati- Tcal 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 about the origins of perfect graphs, I recommend reading the historical 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 K 3,3. 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) w(G') = c(G') for all node-induced subgraphs G' Í G, where w (G) denotes the clique number of G (= largest cardinality of a clique of G, i.e., a set of mutually adjacent nodes) and c(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') = c (G) for all node-induced subgraphs G' Í 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 c (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 Lovász [17], who gave two short and elegant proofs. Lovász [18], in addition, characterized perfect graphs as those graphs G = (V, E) for which the following holds:
(4) w(G') • a (G') ³ |V (G')| for all node-induced subgraphs G' Í G. A link to geometry can be established as follows. Given a graph G = (V, E), we associate with G the vector space RV where each component of a vector of RV is indexed by a node of G.With every subset S Í V, we S S V can associate its incidence vector c = (c v)vÎ V Î R defined by O P T I M A 6 2 JUNE 19 9 9 PAGE 3
S S c v := 1 if v Î S, c v := 0 if v Ï S. The convex hull of all the incidence vectors of stable sets in G is denot- ed by STAB(G), i.e., These three characterizations of perfect graphs provide the link to poly- STAB(G) = conv {c S Î RV | S Í V stable} hedral theory (a graph is perfect iff certain polyhedra are identical) and integer programming (a graph is perfect iff certain LPs have integral solu- and is called the stable set polytope of G. Clearly, a clique and a stable set of tion values). G can have at most one node in common. This observation yields that, Another quite surprising road towards understanding properties of per- for every clique Q Í V, the so-called clique inequality fect graphs was paved by Lovász [19]. He introduced a new geometric x(Q): = å x v £ 1 representation of graphs linking perfectness to convexity and semidefinite v Î Q programming.
is satisfied by every incidence vector of a stable set. Thus, all clique An orthonormal representation of a graph G = (V,E) is a sequence (ui | i V T inequalities are valid for STAB(G). The polytope Î V) of |V| vectors ui Î R such that ||ui||=1 for all i Î V and ui uj = 0 for QSTAB(G) := {x Î RV | 0 £ x " v Î V, x(Q) £1 " cliques Q Í V} all pairs i,j of nonadjacent nodes. For any orthonormal representation v (u | i Î V) of G and any additional vector c of unit length, the so-called called fractional stable set polytope of G, is therefore a polyhedron contain- i orthonormal representation constraint ing STAB(G), and trivially, T 2 V å (c ui ) xi £ 1 STAB(G) = conv {x Î {0,1} |x Î QSTAB(G)}. i Î V Knowing that computing a (G) (and its weighted version) is NP-hard, one V is valid for STAB(G). Taking an orthonormal basis B = {e1, ...,e|V|} of R is tempted to look at the LP relaxation and a clique Q of G, setting c:= ui:=e1 for all i Î Q, and assigning different T max c x, x Î QSTAB(G), vectors of B\{e1} to the remaining nodes i Î V\Q, one observes that every V clique constraint is a special case of this class of infinitely many inequali- where c Î R + is a vector of node weights. However, solving LPs of this type is also NP-hard for general graphs G (see [14]). ties. The set V For the class of perfect graphs G, though, these LPs can be solved in TH(G) := {x Î R + | x satisfies all polynomial time – albeit via an involved detour (see below). orthonormal representation constraints} Let us now look at the following chain of inequalities and equations, is thus a convex set with typical for IP/LP approches to combinatorial problems. Let G = (V,E) be STAB(G) Í TH(G) Í QSTAB(G). some graph and c ³ 0 a vector of node weights: It turns out (see [14]) that a graph G is perfect if and only if any of the
max { å cv | S Í V stable set of G} = following conditions is satisfied: Î v S (8) TH(G) = STAB(G). max {cT x | x Î STAB(G)} = max {cT x | x ³ 0, x(Q) £ 1" cliquesQ Í V , x Î {0,1}V } £ (9) TH(G) = QSTAB(G). T max {c x | x ³ 0, x(Q) £ 1" cliquesQ Í V } = (10) TH(G) is a polytope. min { å yQ | å yQ ³ c v " v Î V , yQ ³ 0" cliquesQ Í V } £ The last result is particularly remarkable. It states that a graph is perfect if Q clique Q' v and only if a certain convex set is a polytope. y y ³ " Î y Î Z " Í min { å Q | å Q c v v V , Q + cliquesQ V } V ' Qclique Q v If c Î R + is a vector of node weights, the optimization problem (with infinitely many linear constraints) The inequalities come from dropping or adding integrality constraints, T max c x, x Î TH(G) 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 can be solved in polynomial time for any graph G. This implies, by (8), follows from (2) that equality holds throughout the whole chain for all that the weighted stable set problem for perfect graphs can be solved in 0/1 vectors c iff G is a perfect graph. This, in turn, is equivalent to polynomial time, and by LP duality, that the weighted clique covering problem, and by complementation, that the weighted clique and coloring (5) The value max {cTx | x Î QSTAB(G)} problem can be solved in polynomial time. These results rest on the fact is integral for all c Î {0,1}V. that the value Results of Fulkerson [10] and Lovász [17] imply that (5) is in fact equiva- T J (G, c) := max {c x | x Î TH(G)} lent to can be characterized in many equivalent ways, e.g., as the optimum value (6) The value max {cTx | x Î QSTAB(G)} is integral for all c Î ZV . + of a semidefinite program, the largest eigenvalue of a certain set of sym- and that, for perfect graphs, equality holds throughout the above chain for metric matrices, or the maximum value of some function involving V all c Î Z +. This, as a side remark, proves that the constraint system defin- orthornormal representations. ing QSTAB(G) in totally dual integral for perfect graphs G. Chvátal [6] Details of this theory can be found, e.g., in Chapter 9 of [14]. The observed that (6) holds iff algorithmic results involve the ellipsoid method. It would be nice to have STAB(G) = QSTAB(G) “more combinatorial” algorithms that solve the four optimization prob- lems for perfect graphs in polynomial time. O P T I M A 6 2 JUNE 19 9 9 4
Let us now move into information theory. Given a graph G = (V,E), we • |V| = a (G) • w (G) + 1, V call a vector p Î R + a probability distribution on V if its components sum • G has exactly |V| maximum cliques, and every node is contained in to 1. Let G(n) = (Vn, E(n)) denote the so-called n-th conormal power of G, exactly w (G) such cliques. n i.e., V is the set of all n-vectors x = (x1,…, xn) with components xi Î V, • G has exactly |V| maximum stable sets, and every node is contained and in exactly a (G) such stable sets. (n) n • QSTAB(G) has exactly one fractional vertex, namely the point E := {xy | x, y Î V and $ i with xi yi Î E} x = 1/w (G) " v Î V, which is contained in exactly |V| facets and Each probability distribution p on V induces a probability distribution pn v n n adjacent to exactly |V| vertices, the incidence vectors of the maximum on V as follows: p (x) : = p (x ) • p(x ) • … • p(x ). For any node set U Í i 2 n stable sets. Vn, let G(n)[U] denote the subgraph of G(n) induced by U and C (G(n) [U]) Similar investigations (but not resulting in such strong structural its chromatic number. Then one can show that, for every results) have recently been made by Annegret Wagler [24] on graphs 0 < e < 1, the limit 1 which are perfect and have the property that deletion (or addition) of any H(G, p): = lim min log C(G (n) [U ]) n ®¥ n p n(U )³ 1-e 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 exists and is independent of e (the logs are taken to base 2). H(G,p) is edge is added to G or any edge is deleted from G the resulting graph is called the graph entropy of the graph G with respect to the probability imperfect. distribution p. If G = (V,E) is the complete graph, we get the well-known Particular efforts have been made to characterize perfect graphs “con- Shannon entropy structively” in the following sense. One first establishes that a certain class
H(p) = - å pi log pi. C¥ of graphs is perfect and considers, in addition, a finite list CÎ of special iÎ V perfect graphs. Then one defines a set of “operations” (e.g., replacing a Let us call a graph G = (V,E) strongly splitting if for every probability dis- node by a stable set or a perfect graph) and “compositions” (e.g., take two tribution p on V graphs G and H and two nodes u Î V (G) and v Î V(H), define V (G ° H) – H(p) = H (G,p) + H (G, p) : = (V(G) È V(H)) \{u,v} and E (G ° H) : = E (G – u) È E (G – v) È {x,y | xu Î E (G), yv Î E (H)} and shows that every perfect graph can be con- holds. Csiszár et. al [9] have shown that a graph is perfect if and only if structed from the basic classes C¥ and CÎ by a sequence of operations and G is strongly splitting. compositions. Despite ingenious constructions (that were very helpful in I.e., G is perfect iff, for every probabiltity distribution, the entropies of proving some of the results mentioned above) and lots of efforts, this route – G and of its complement G add up to the entropy of the complete graph of research has not led to success yet. A paper describing many composi- (the Shannon entropy). I recommend [9] for the study of graph entropy tions that construct perfect graphs from perfect graphs is, e.g., [8]. and related topics. Chvátal [7] initiated research into another “secondary structure” related Given all these beautiful characterizations of perfect graphs and polyno- to perfect graphs in order to come up with a (polynomial time recogniza- mial time algorithms for many otherwise hard combinatorial optimization ble) certificate of perfection. For a given graph G = (V,E), its P4-structure is problems, it is really astonishing that nobody knows to date whether per- the 4-uniform hypergraph on V whose hyperedges are all the 4-element fectness of a graph can be recognized in polynomial time. There are many node sets of V that induce a P4 (path on four nodes) of G. Chvátal ways to prove that, deciding whether a graph is not perfect, is in NP. But observed that any graph whose P4-structure is that of an odd hole is an that’s all we know! odd hole or its complement and, thus, conjectured that perfection of a Many researchers hope that a proof of the most famous open problem graph depends solely on its P4-structure. Reed [23] solved Chvátal’s semi- in perfect graph theory, the strong perfect graph conjecture: strong perfect graph conjecture by showing that a graph G is perfect iff A graph G is perfect if and only if G neither contains an odd hole (12) G has the P4-structure of a perfect graph. nor an odd antihole as an induced subgraph. There are other such concepts, e.g., the partner-structure, that have results in structural insights that lead to a polynomial time algorithm for resulted in further characterizations of perfect graphs through secondary recognizing perfect graphs. It is trivial that every odd hole (= chordless structures. We recommend [15] for a thorough investigation of this topic. cycle of length at least five) and every odd antihole (= complement of an But the polynomial-time-recognition problem for perfect graphs is still odd hole) are not perfect. Whenever Claude Berge encountered an imper- open. fect graph G he discovered that G contains an odd hole or an odd antihole A relatively recent line of research in the area of structural perfect graph and, thus, came to the strong perfect graph conjecture. In his honor, it is theory is the use of the probability theory. I would like to mention just customary to call graphs without odd holes and odd antiholes Berge one nice result of Prömel und Steger [22]. Let us denote the number of graphs. Hence, the strong perfect graph conjecture essentially reads: every perfect graphs on n nodes by Perfect (n) and the number of Berge graphs Berge graph is perfect. on n nodes by Berge (n), then This conjecture stimulated a lot of research resulting in fascinating Perfect (n) lim =1. insights into the structure of graphs that are in some sense nearly perfect n®¥ Berge (n) or imperfect. E.g., Padberg [20], [21] (introducing perfect matrices and In other words, almost all Berge graphs are perfect, which means that if using proof techniques from linear algebra) showed that, for an imperfect there are counterexamples to the strong perfect graph conjecture, they are graph G = (V,E) with the property that the deletion of any node results in “very rare.” a perfect graph, satisfies the following: O P T I M A 6 2 JUNE 19 9 9 PAGE 5
The theory of random graphs provides deep insights into the proba- References [13] M. C. Golumbic, Algorithmic bilistic behavior of graph parameters (see [4], for instance). To take a sim- graph theory and perfect graphs, ple example, consider a random graph G = (V,E) on n nodes where each [1] C. Berge, The history of the Academic Press, New York, edge is chosen with probality ½. It is well known that the expected values 1980. – perfect graphs, Southeast Asian of a (G) and w(G) are of order log n while C(G) and C(G) both have Bull. Math. 20, No.1 (1996) [14] M. Grötschel, L. Lovász, A. expected values of order n/ log n. This implies that such random graphs 5-10. Schrijver, Geometric algorithms are almost surely not perfect. An interesting question is to see whether the [2] C. Berge, Motivations and and combinatorial “LP-relaxation of a (G),” the so-called fractional stability number a * (G) = history of some of my conjec- optimization, Springer, Berlin- max {1T x | x Î QSTAB(G)}, is a good approximation of a (G). Observing tures, Discrete Mathematics Heidelberg, 1988. [15] S. Hougardy, On the P -struc- that the point x = (x ) with x : = 1 /w(G), v Î V, satisfies all clique 165-166 (1997) 61-70. 4 v vÎ V v ture of perfect graphs, Shaker- constraints and is thus in QSTAB(G) and knowing that w (G) is of order [3] C. Berge, V. Chvátal (eds.), Verlag, Aachen 1996. log n one can deduce that the expected value of a *(G) is of order n /log n, Topics on perfect graphs, – [16] S. Hougardy, Inclusions i.e., it is much closer to C(G) than to a (G). Hence, somewhat surprising- Annals of Discrete Mathematics 21, North-Holland, Between Classes of Perfect ly, a *(G) is a pretty bad approximation of a (G) in general – not so for Amsterdam, 1984. Graphs, Preprint, Institut für perfect graphs, though. [4] B. Bollobás, Random Graphs, Informatik, Humboldt- To summarize this quick tour through perfect graph theory (omitting Academic Press, 1985. Universität zu Berlin, 1998, quite a number of the other interesting developments and important [5] A. Brandstädt, V. B. Le, J. P. see results), here is my favorite theorem: Spinrad, Graph Classes: A
solutions to the problems to Robert
Bosch ([email protected]). The
most attractive solutions will be pre-
sented in a forthcoming issue.
The 8-queens problem requires that eight Problems Peaceably Coexisting queens be placed on a chessboard in such a 1. Figure 2 demonstrates that two armies of Armies of Que e n s way that no two can attack each other (i.e., lie eight queens can peaceably coexist on a in the same row, column, or diagonal). See chessboard (i.e., be placed on the board Figure 1 for a solution. The 8-queens problem in such a way that no two queens from Robert A. Bosch was proposed by a chessplayer named Max opposing armies can attack each other). Bezzel in 1848. During the next two years, it Formulate an integer program to find the June 14, 1999 captured the attention of Carl Friedrich Gauss maximum size of two equal-sized, peace- (albeit briefly) and other prominent scholars. ably coexisting armies of queens. 2. How many optimal solutions does the In September 1850, Franz Nauck presented 92 integer program have? solutions. See [2] for a detailed history. Please send solutions and/or comments to
Pentomino Exclusion Revisited In the December 1998 issue of OPTIMA, we presented an integer programming formula- tion of the n x n pentomino Figure 1 Figure 2 exclusion problem. This prob- lem requires that monominoes Today, the problem continues to fascinate. (1 x 1 square) be placed on an n x n board in A web page maintained by Walter Kosters lists such a way that there is no room left for any 66 recent books and articles that refer to the pentominoes (objects formed by joining five 8-queens problem or its generalization, the n- 1 x 1 squares in an edge-to-edge fashion). The queens problem [5]. One article, by L.R. goal is to use as few monominoes as possible. Foulds and D.G. Johnston, describes two We also challenged readers to find a better strategies for solving various “chessboard non- formulation, and we were pleased to receive a attacking puzzles”: maximal cliques and inte- couple of excellent submissions. One, by ger programming [3]. Frank Plastria, contains a description of several Here, we pose a variant. families of valid inequalities for our original formulation as well as proofs that some of these inequalities define facets [6]. The second, O P T I M A 6 2 JUNE 19 9 9 PAGE 7
by Kurt Anstreicher, does not discuss the formu- empty squares as well as portions of neighboring lation at all. Instead, it gives a simple, direct squares (and portions of monominoes). proof that at least 24 monominoes are needed to The lower lefthand corner of Figure 5 displays exclude all pentominoes from an 8 x 8 board the modified board. We created it by expanding [1]. the original board (by adding a border of partial In the next section, we discuss Plastria’s valid squares) and then “triangulating” the expansion inequalites. And then, in the three sections that (by dividing each square and partial square into follow, we describe a completely different formu- equal-sized triangular pieces). lation of the problem, present evidence that it is The rest of Figure 5 displays each tile in its vastly superior to the original formulation, and “standard orientation.” Note that there is a tile show how its constraints can be used to con- for each possible clump of empty squares. Each struct a simple proof of a result concerning Figure 4 of tiles 1 and 5 has only one orientation (hence- toroidal pentomino exclusion problems. forth referred to as “number 1”). Each of tiles 2, 3, and 6 has two: its standard orientation (num- Valid Inequalities for the Original ber 1), and the orientation obtained by rotating Formulation the standard orientation 90 degrees clockwise (number 2). Similarly, each of tiles 4 and 9 has In [6], Plastria begins by noting that the origi- four orientations: its standard orientation (num- nal formulation can be written very concisely: ber 1), and the orientations obtained by rotating
minimize å xs the standard orientation 90, 180, and 270 sÎ S degrees clockwise (numbers 2, 3, and 4). Tile 7 subject to å xs ³ 1 for all P Î P has eight orientations, and tile 8 has four. In sÎ P
xs Î {0,1} for all s Î S. each of these last two cases, half of the orienta- tions are reflections (orientations that result from Here, S stands for the set of all squares of the “picking up” another orientation and “flipping it board, and P stands for the set of all pentomino over”). placements. Plastria considers a pentomino Figure 6 demonstrates how some of these tiles placement to be a 5-element subset of S that can be assembled to form the solution presented consists of the five squares that are covered in [4]. when some pentomino is placed – either in its We now present the details of the set packing “standard” orientation or in some rotated or formulation. We begin by defining a set reflected orientation – somewhere on the A = {(t,o,i,j): tile t lies entirely on the Figure 5 board. board if placed there in orientation o with Plastria goes on to construct several families of A Completely Different Formulation is reference point (the small circle) in the “hexomino” constraints. He proves that his hex- center of square i,j} omino constraints are valid inequalities and that Figure 4 displays an optimal some are facet-generating. Figure 3 contains solution to the 8 x 8 pen- graphical displays of three families of hexomino tomino exclusion problem. constraints. All three are facet-generating. The Note that in the solution, one on the left states that at least two of the six empty squares (squares with- squares in any 2 x 3 rectangular-shaped region out monominoes) appear in should receive monominoes. The center one “clumps.” In fact, each clump states that if the central square of a region that of empty squares is a poly- forms a “ribbon-shaped” hexomino does not omino (“skew” tetrominoes, receive a monomino, then at least two of the T tetrominoes, and L tetro- other five squares of the region must receive minoes). monominoes. This simple fact can serve Figure 6 as the foundation of a set packing formulation and, for each triangular piece t of the board, a of polyomino exclusion problems. (The original set formulation is a set covering formulation.) Here, C = {(t,o,i,j)Î A : tile t covers triangular Figure 3 we form solutions not by placing individual t piece t if placed on the board in orientation monominoes on individual squares of the board, o with is reference point in the center of as in the original, set covering approach, but by square i,j} placing “tiles” on a modified version of the board. Each tile consists of an entire clump of O P T I M A 6 2 JUNE 19 9 9 8
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 Ct contains all admissible place-
ments that cover triangular piece t.We call Ct 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)Î A,
ïì 1 if admissible placement (t, o,i, j ) is used, yt,o ,i, j = í îï 0 otherwise.
And the set packing formulation contains one Figure 7 Figure 8 constraint for each triangular piece t of the board: mulation is far superior to the original, set cover-
å yt, o, i, j £ 1. ing formulation. The set packing formulation mino tiles (tile 9 in Figure 5). (t, o,i, j) Î C t was able to solve many more of the problems We conclude this section with a theorem These constraints make sure that each trian- tested, and it required considerably fewer about toroidal pentomino exclusion problems. gular piece of the board is covered by at most branch-and-bound nodes on all problems tested, Theorem: For all n, all solutions to the n x n one tile. considerably fewer iterations of the simplex toroidal pentomino exclusion problem have den- We minimize the number of monominoes method on all but one of the problems tested sity greater than or equal to 3/7. placed on the board by maximizing the total (the smallest, the 6 x 6 problem), and much less Proof: First, add up all constraints of the set number of empty squares on the board. If, for 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 each tile t, vt denotes the number of empty into 8n2 triangular pieces, the resulting inequali- squares contained within (note that v1 = 1, v2 = ty is 2, v3 = v4 = 3, and v5 = v6 = • • • = v9 = 4) then Toroidal Problems
our objective is to maximize Figures 7 and 8 display two optimal solutions to 2 (1) å å yt,o ,i , j £ 8n . å vt yt ,o ,i, j. the 14 x 14 toroidal pentomino exclusion prob- t (t,o, i, j)Î C t (t, o,i, j)Î A lem. (An n x n toroidal board is constructed by gluing together both pairs of opposite borders of (The first summation is over all triangular pieces a standard n x n board.) Note that each solution t .) Now let d denote the number of triangular Comparing the Formulations t has a density of 3/7. (We consider the density of pieces in tile t. Note that d = 16, d = 32, d = We tested the two formulations on a number of 1 2 3 a solution to be the fraction of squares of the 48, d = 44, d = 56, d = 64, d = 60, and d = d n x n pentomino exclusion problems, using a 4 5 6 7 8 9 board that have monominoes in them.) Also = 56. Furthermore, note that 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 (2) å å y = å dt y . Table 1 makes it clear that the set packing for- t,o ,i , j t,o ,i. j the Figure 8 solution is made entirely of T tetro- t (t,o, i, j)Î C t (t,o ,i, j) Î A
(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 O P T I M A 6 2 JUNE 19 9 9 PAGE 9
d 8n 2 4n 2 Using this notation, we can write Plastria’s t £ = å y t,o,i , j . formulation as follows: (t,o,i , j )Î A 14 14 7
min å y p,o ,i, j ( p,o ,i, j) Î A s And since d/14 ³ vt for each tile t, it fol- £ £ £ lows that s.t. å y p,o,i ', j' 1 1 i , j n 2 ( p,o ,i', j') Î C s 4n i , j å vt yt, o,i, j £ (t, o,i, j)Î A 7 å y p,o,i , j £ 1 1 £ p £ 12 o,i , j: ( p,o ,i, j) Î A s Hence the density is greater than or equal to s å y p,o', i', j' + å y p', o',i', j' £ 1 (p,o,i, j) Î A 3/7. o',i', j': ( p',o',i', j') Î S p , o ,i , j s (p,o ',i ', j')Î A s The Pentomino Spanning Problem y p ,o,i , j Î {0,1} (p,o,i, j) Î A In the December 1998 issue of OPTIMA, we also challenged readers to devise an integer pro- The variables are basically the same as in the set gramming formulation for finding the smallest packing formulation of the pentomino exclu- S subset of pentominoes that spans an n x n sion problem: for each (p,o,i,j)Î A , board. (A subset of pentominoes spans a board ïì 1 if admissible placement (p ,o,i, j ) is used, if its members can be placed on the board in y p,o ,i, j = í îï 0 otherwise. such a way that they exclude the remaining pentominoes.) In this section, we present a for- The first set of constraints ensures that each mulation due to Frank Plastria [6]. For the sake square is covered by at most one pentomino. of brevity, instead of Plastria’s notation we use The second set guarantees that each pentomino notation very similar to that used in the previ- is placed on the board at most once. The third ous three sections. set makes sure that the pentominoes that are We begin by numbering the pentominoes, placed on the board actually span the board. In numbering the orientations of each pentomino, particular, the constraint corresponding to and selecting one square of each pentomino to admissible placement (p,o,i,j) makes sure that if serve as that pentomino’s reference square. We pentomino p is not placed on the board, then at then define a set of admissible placements least one of the squares that would have been covered by (p,o,i,j) is covered. To see this, note AS = {(p,o,i,j) : pentomino p lies entirely that on the board if placed there in orientation 1. the leftmost summation takes on a value of o with its reference square on square i,j} 0 if and only if pentomino p is no t placed and, for each square i,j of the board, a covering on the board, and set 2. the rightmost summation takes on a posi- S S tive value if and only if at least one of the C i,j = {(p,o,i',j') Î A : pentomino p covers square i,j, if place on the board in orienta- squares that would have been covered by tion o with its reference square on square (p,o,i,j) is covered. i',j'}. In addition, for each admissible placement (p,o,i,j) Î AS, we define a set S Sp,o,i,j = {(p',o',i',j') Î A : (p',o',i',j') covers at least one of the squares covered by (p,o,i,j)}. 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,
Co n fe r e n c e
Atl a n t a
Second International Workshop on Approximation Algorithms for Combinatorial Optimization Probl e m s , and Th i r d International Workshop on Randomization and Approximation Tec hniques in Computer Science August 8-11, 19 9 9 ,B e r k e l e y, C A ,U S A ht t p : / / c u i w w w. u n i g e.ch / ~ ro l i m / a p p r ox;h t t p : / / c u i w w w. u n i g e.ch / ~ ro l i m / r a n d o m Symposium on Operations Research 1999, SOR ‘9 9 September 1-3, 19 9 9 ,M ag d e bu rg ,G e r m a n y ht t p : / / w w w. u n i - m ag d e bu rg . d e / S O R 9 9 / Sixth International Conference on Parametric Optimization and Related Top i c s October 4-8, 1 9 9 9 ,D u b rovn i k ,C ro a t i a ht t p : / / w w w. m a t h . h r / d u b r ovn i k / i n d ex . h t m INFORMS National Meeting No vember 7-10, 19 9 9 ,P h i l a d e l p h i a , PA ,U S A ht t p : / / w w w. i n fo r m s . o rg / C o n f / P h i l a d e l p h i a 9 9 / 7th INFORMS Computing Society Conference on Computing and Optimization:Tools for the New Millenium Ja nu a r y 5-7, 20 0 0 ,C a n c u n ,M ex i c o ht t p : / / w w w - bu s . c o l o r a d o . e d u / Fa c u l t y / L ag u n a / c a n c u n 2 0 0 0 . h t m l DI M A CS 7th Implementation Challenge :S e m i d e finite and Related Optimization Prob lems Wor k s h o p Ja nu a r y 24-26, 20 0 0 ,R u t ge r s Universi t y ;P i s c a t aw a y, NJ ht t p : / / d i m a c s . r u t ge rs . e d u / Wo r k s h o p s / 7 t h ch a l l e n g e Se venth International Workshop on Project Management and Scheduling (PMS 2000) April 17-19, 20 0 0 ,U n i v e r sity of Osnabrueck ,G e r m a ny ht t p : / / w w w. m a t h e m a t i k . u n i -os n a b r u e ck . d e / r e s e a r ch/ O R / p m s 2 0 0 0 / Applied Mathematical Programming and Modelling Conference (APMOD 2000) 17-19 April 2000, Brunel Universi t y , Lo n d o n ht t p : / / w w w. a p m o d . o rg . u k ISMP 2000 17th International Symposium on Mathematical Prog r a m m i n g August 7-11, 20 0 0 ,G e o r gia Institute of Tech n o l o g y , A t l a n t a ,G A ,U S A ht t p : / / w w w. i s y e.g a t e ch . e d u / i s m p 2 0 0 0 O P T I M A 6 2 JUNE 19 9 9 PAGE 11
Fi r st Announcement The 17th International Symposium on Mathematical Programming General Infor m a t i o n Call For Pap e r s August 7-11, 2000 Georgia Institute of Technology, Atlanta, Georgia, USA.
http://www.isye.gatech.edu/ismp2000
Pap e r s on all theoretical, co m p u t a t i o n a l standing papers in discrete mathematics); List of Top i c s and practical aspects of mathematical Orchard-Hays Prize (for excellence in Sessions on the following topics are planned. pr ogramming are welcome.The presen- computational mathematical program- Suggestions for further areas to be included are welcome. tation of very recent results is encour- ming); and A.W.Tucker Prize (for an (A) Approximation Algorithms, aged . outstanding paper by a student). (B) Combinatorial Optimization, Morning Historical Pers p e c t i v e s (C) Complementarity and Variational Inequalities, De a d l i n e s March 31, 2000: deadline for Pl e n a r i e s Nonlinear Programming: (D) Computational Biology, submission of titles and abstracts, and Roger Fletcher; Integer Programming: (E) Computational Complexity, deadline for early registration. Martin Grötschel; Combinatorial (F) Computational Geometry, Optimization: William Pulleyblank; (G) Convex Programming and Nonsmooth Optimization, (H) Dynamic Programming and Optimal Control, Si t e The symposium will take place at Linear Programming: Michael Todd; the Georgia Institute of Technology (I) Finance and Economics, Developments in the Soviet Union: located in Atlanta, Georgia, USA. Hotels (J) Game Theory, Boris Polyak. of various categories will be available as (K) Global Optimization, Afternoon Semi-Plenaries Aharon well as low priced accommodations on (L) Graphs and Networks, Ben-Tal, William Cunningham, the Georgia Tech campus. (M) Integer and Mixed Integer Programming, Don Goldfarb, Alan King, Jim Orlin, (N) Interior Point Algorithms and Semi-Definite Programming, James Renegar, Alexander Schrijver, Or ganizing Committee (O) Linear Programming, David Williamson, Yinyue Ye. General Co-Chairs: G.L. Nemhauser, (P) Logistics and Transportation, M.W.P. Savelsbergh (Q) Multicriteria Optimization, Program Committee Co-Chairs: Social Prog r a m (R) Nonlinear Programming, E.L. Johnson, A. Shapiro, J. VandeVate, Sunday, August 6: Welcome reception, (S) Parallel Computing, R. Monteiro, R. Thomas, C. Tovey, 18:00 – 19:30 (T) Production Planning and Manufacturing, V. Vazarani; Monday, August 7: Opening ceremony, (U) Scheduling, Local Committee Chair: D. Llewellyn; 9:00 – 12:00 (V) Semi-Infinite and Infinite Dimensional Programming, Committee Members: F. Al-Khayyal, Wednesday, August 9: Evening reception, (W) Software, E. Barnes, P. Keskinocak, A. Kleywegt, 18:00 – 23:00 (X) Statistics, J. Sokol A special program for accompanying (Y) Stochastic Programming, and Structure of the Meeting A large number persons is being organized. (Z) Telecommunications and Network Design. of people will be invited to organize ses- sions. Others may nominate themselves Im p o r tant Addr e ss e s as session organizers. To do so, they must Mailing address: ISMP 2000, c/o A. write us and, upon our program com- Race, School of Industrial and Systems mittee’s aproval, they will be included in Engineering, Georgia Institute of Call for Proposals to Hos t the list. Technology, Atlanta, GA 30332-0205, During the plenary opening session, the USA; I S M P 2 O O 3 following prizes will be awarded: Fax: (+1) 404 894 0390; Dantzig Prize (for original research hav- E-Mail:
Workshop on Discrete Venu e The DO’99 workshop will Workshop on the Th e o r y and Practice of Integer Programming Optimization (DO’99) take place on the Busch Campus of in Honor of Ralph E. Gomory DIMACS/RUTCOR, Rutgers Rutgers, The State University of on the Occasion of his 70th Birthday University New Jersey, July 25-30, 1999.
July 25-30, 1999 In vited Surve y Talks Egon Balas http://rutcor.rutgers.edu/~do99 (Carnegie Mellon University), Lift- e are pleased to announce a workshop in celebration of Ralph and-project: progress and some The updated announcement below Gomory’s 70th birthday. The focus of the workshop will be on open questions; Peter Brucker includes, besides the information integer linear programming. The workshop is sponsored by (University of Osnabrueck), W contained in the first announce- Complex scheduling problems; DIMACS, as part of the 1998-99 Special Year on Large-Scale Discrete ment, new information concerning: Rainer Burkard (Technical Optimization, and by IBM. The workshop will be held August 2-4, 1999, 1. Registration (including dead- University of Graz), Assignment at the IBM Watson Research Center in Yorktown Heights, New York. The lines); 2. Support for participants problems; Vasek Chvatal (Rutgers workshop will include lectures by leading international experts covering all (including the availability of an University), On the solution of aspects of integer programming. We hope that the lecture program will be NSF grant); 3. Accommodations traveling salesman problems; of particular interest to young researchers in the field, including Ph.D. stu- (including housing request dead- Gerard Cornuejols (Carnegie Mellon lines); and 4. Publications. dents and post-doctoral fellows. University), Packing and covering; A conference banquet will be held with Alan Hoffman (IBM) as the Master Yves Crama (University of Liege), Go a l s Discrete optimization under- of Ceremonies. The banquet speakers will include Paul Gilmore (University Optimization models in produc- went a tumultuous development in of British Columbia), Ellis Johnson (Georgia Tech), and Herb Scarf (Yale). tion planning; Fred Glover the last half century and had a par- In vited Lecturers inclu d e : Karen I. Aardal, Utrecht University; Egon Balas, (University of Colorado), Tabu ticularly spectacular growth in the search & evolutionary methods: Carnegie Mellon University; Francisco Barahona, IBM Watson Research last few decades. The main goal of unexpected developments; Alan Center; Imre Barany, Hungarian Academy of Sciences; Daniel Bienstock, this workshop is to survey the state Hoffman (IBM Research Center), Columbia University; Robert Bixby, Rice University; Charles E. Blair, of the art in discrete optimization. Greedy algorithms in linear pro- University of Illinois; Vasek Chvatal, Rutgers University; Sebastian Ceria, This goal will be achieved by the gramming problems; Karla Columbia University; Gerard Cornuéjols, Carnegie Mellon University; presentation of expository lectures Hoffman (George Mason William H. Cunningham, University of Waterloo; John J. Forrest, IBM presenting the major subareas of University), Applications of col- the field, including its theoretical Watson Research Center; Michel X. Goemans, Université Catholique de umn-generation and constraint- foundations, its methodology and Louvain; Ralph Gomory, Sloan Foundation; Peter Hammer, Rutgers generation methods; Toshihide applications. The surveys will be University; T.C. Hu, University of California at San Diego; Ellis Johnson, Ibaraki (Kyoto University), Graph presented by some of the most Georgia Tech; Mike Juenger, Universitat zu Koeln; Berhard Korte, connectivity & its augmentation; prominent researchers in the field. University of Bonn; Thomas L. Magnanti, Massachusetts Institute of Bernhard Korte (University of DO’99 will also provide a forum Bonn), Gigahertz-processors need Technology; George L. Nemhauser, Georgia Institute of Technology; Gerd for the presentation of new devel- discrete optimization; Jakob Krarup Reinelt, Universität Heidelberg; Martin W.P. Savelsbergh, Georgia Institute opments in discrete optimization. (University of Copenhagen), of Technology; Herbert E. Scarf,Yale University; Andras Sebö, University of To accomplish this goal, DO’99 Locational decisions with friendly Grenoble; Bruce Shepherd, Lucent Bell Laboratories; Bernd Sturmfels, will feature a series of sessions for and obnoxious facilities; Tom University of California at Berkeley; Mike Trick, Carnegie Mellon the presentation of contributed Magnanti (M.I.T.), Network University; Leslie Earl Trotter, Jr., Cornell University; Robert Weismantel, talks, presenting the latest research design; Silvano Martello (University of the participants. University of Magdeburg; David P.Williamson, IBM Watson Research of Bologna), Bin packing problems DO ’ 9 9 is being held 22 years after Laboratory; Laurence Alexander Wolsey, Université Catholique de Louvain; in two and three dimensions; DO’77, which had very similar and Günter Ziegler,Technische Universität Berlin. George L. Nemhauser (Georgia goals, and whose collection of sur- Co n f erence Orga n i ze rs : William Cook, Rice University; and William Institute of Technology), Discrete veys (Discrete Optimization I and Pulleyblank, IBM Watson Research Center. Optimization in Air II, Annals of Discrete Mathematics, Transportation; Jim Orlin and Ravi For more details, please see vols. 4 and 5, P.L. Hammer, E.L. Ahuja (M.I.T.), Neighborhood
DI M A CS 7th Implementation Challenge: APMOD 2000 Se m i d e fi nite and Related Optimization Prob lems Wor k s h o p The Applied Mathematical Prog r a m m i n g DIMACS Center, CoRE Building, Rutgers University; Piscataway, NJ and Modelling Conference (APMOD 2000) January 24-26, 2000 Brunel University, London, 17-19 April 2000 Org a n i ze r s Farid Alizadeh, RUTCOR, Rutgers ting plane methods, or branch and bound; truss The objective of APMOD 2000 is to bring University; David Johnson, AT&T Labs – topology, and Steiner tree problems lacking a together active researchers, research students Research; Gabor Pataki, Columbia University strictly complementary solution. and practitioners from various countries and Presented under the auspices of the Special We invite papers dealing with all computational provide a forum for discussing and presenting Year on Large Scale Discrete Optimization. aspects of semidefinite programming and related established as well as new techniques in The purpose of DIMACS computational chal- problems. In particular, the following classes of Operational Research/Management Science. lenges has been to encourage the experimental problems are of special interest: (1) Cut, and APMOD 2000 provides a bridge between evaluation of algorithms, in particular those with partition problems; (2) Theta function, and emerging research and their applicability in efficient performance from a theoretical point of graph entropy problems; (3) SDP relaxation of industry. The theme of APMOD 2000 is view. The past Challenges brought together very difficult, (currently unsolvable) 0-1 mixed chosen to be Corporate Application of researchers to test time-proven, mature, and integer programming problems; (4) Problems in Mathematical Optimisation. The conference novel experimental approaches on a variety of convex quadratically constrained quadratic pro- embraces a broad range of classical OR topics problems in a given subject. As the subject of grams from engineering; (5) Problems from sta- including linear programming, integer pro- the last Challenge of this century, one could tistics and finance; (6) SDP instances from engi- gramming, combinatorial optimisation, sto- hardly think of a better choice than Semidefinite neering, for example truss topology design, and chastic programming and nonlinear program- Programming (SDP), one of the most interest- control theory problems; (7) Difficult, randomly ming. Emphasis is placed on novel techniques ing and challenging areas in optimization theory generated problems designed to challenge algo- of approaching these problems and their to emerge in the last decade. In the past few rithms on performance and numerical stability; application in industries such as finance, the years, much has been learned on both the kinds (8) In addition to the classes above, we invite utilities, transport and many other diverse of problem classes that SDP can tackle, and the investigations which focus on using SDP and areas of interest. best SDP algorithms for the various classes. In related codes to test out behavior of heuristics The three-day event will have plenary sessions addition, a great deal has been learned about the and other new applications. The SDP code can by leading researchers and also four parallel limits of the current approaches to solving be developed by the investigators or they may streams. The written contributions will be SDP’s. choose off-the-shelf codes. refereed and published in the Annals of A closely related problem to semidefinite pro- All communications regarding the challenge Operations Research. gramming is that of convex quadratically con- should be directed to
Following the six successful work- (Poznan University of Technology), shops in Lisbon (Portugal), Como Luis Valadares Tavares (Instituto (Italy), Compiegne (France), Superior Technico, Lisbon), Leuven (Belgium), Poznan Gunduz Ulusoy (Bogazici (Poland), and in Istanbul (Turkey), University, Istanbul), Vicente Valls the Seventh International (University of Valencia), Jan Workshop on Project Management Weglarz (Poznan University of and Scheduling is to be held in Technology), and Robert J. Willis Osnabrück, a small, charming city (Monash University). located halfway between Cologne and Hamburg. Pr e r e g i s t r a t i o n If you are interest- The main objectives of PMS 2000 ed in participating, please visit our are to bring together researchers in web site the area of project management
Linear Semi-Infinite Optimization Graph Theory Miguel A. Goberna and Marco A. Lopez Reinhard Diestel John Wiley & Sons, 1998 Springer Verlag
ISBN 0-471-97040-9 ISBN 0-387-98210-8
inear semi-infinite prog r a m m i n g deals with the problem of minimiz- ave you heard of Szemerédi’s regularity lemma? If you have been in ing (maximizing) a linear objective function of a finite number of touch with graph theory at least a little, the answer is almost sure- variables with respect to an (possibly and generally) infinite num- ly yes. How about its proof? In case you were so far too afraid to ber of linear constraints. There is a great variety of applications of go through the details and understand the basic ideas of this far- L semi-infinite optimization, including problems in approximation reaching result, the book Graph Theory by G. Diestel will prove an theory (using polyhedral norms), operation research, optimal control, H excellent guide. One of the main strengths of this book is making deep and boundary value problems and others. These applications and appealing difficult results of graph theory accessible. theoretical properties of semi-infinite problems gave rise to intensive (and Actually, the goal of this book is (at least) twofold: First, to give a solid up to now undiminished) research activities in this field since their incep- introduction to basic notions and provide the standard material of graph tion in the 1960s. theory, such as the matching theorems of König, Hall, Tutte, Petersen, the The book under review is, according to the authors, intended “... as a theorems of Dilworth and Menger on paths and chains, Kuratowski’s char- monograph as well as a textbook …” It fills a gap in the present literature acterization of planar graphs, coloring theorems of Brooks and Vizing, etc. about optimization. But already the introductory chapters include results which have not yet The authors organize their material into four parts: appeared in textbooks: Mader’s theorem (1.4.2) on the existence of k-con- Part I, Modeling, deals with numerous examples of occurring linear nected subgraphs of a sufficiently dense graph, Tutte’s characterization of 3- s e m i - i n finite optimization problems, divided into two chapters (1) connected graphs. It is also refreshing to see the proof of Mader’s theorem Modeling with the primal problem and (2) Modeling with the dual (3.6.1) on the existence of large topological complete graph as a minor in problem. Most of the models described in Part I arise from other fields of a dense graph, or of the pretty theorem of Jung and of Larman and Mani applied mathematics. (3.6.2) stating the existence of k disjoint paths between any set of k pairs of In Part II, Linear Semi-Infinite Systems, the feasible sets of (primal) nodes in a sufficiently highly connected graph. linear semi-infinite optimization problems are investigated. It provides the The main novelty of this book, however, is that a great number of diffi- necessary fundamentals for the forthcoming theory and contains the cult and deep theorems, along with their full proofs, are exhibited. These chapters (3) Alternative Systems, (4) Consistency, (5) Geometry and (6) include some older results like Szemerédi’s above-mentioned regularity Stability. 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 O P T I M A 6 2 JUNE 19 9 9 16
The essential Part III, Theory of Linear Semi-Infinite Programming, (Theorem 9.3.1) (due to three groups of authors around the beginning of presents in chapter (7) Optimality a natural extension of the classical the ‘70s) from induced Ramsey theory is also completely proved (four Karush-Kuhn-Tucker theory to linear semi-infinite optimization problems pages). The classical results on algebraic flows, including Tutte’s investiga- and provides further information on some optimality conditions for the tions and Seymour’s 6-flow theorem, are fully covered as well. Not only dual problem and for convex semi-infinite programming problems. Lovász’s perfect graph theorem (5.5.3), stating the perfectness of the com- Chapter (8), Duality, develops a systematic approach to the main topics plement of a perfect graph, is discussed, but his second, significantly deep- of duality theory in linear semi-infinite programming. Additionally, the er characterization of perfect graphs, as well (Theorem 5.5.5). (Naturally, (occasionally) occurring duality gaps are analyzed, as well as the connection not every fundamental result of graph theory could be included with its between duality gaps and the applicability of discretization methods. proof; but perhaps – if the reviewer may make a suggestion – the author Chapter (9), Extremality and Boundedness, deals with extreme points may find a way in the next edition to exhibit a proof of Mader’s beautiful and extreme directions of the feasible sets and optimal sets of a (fixed) pair theorem (3.4.1) on the maximum number of independent H-paths.) of linear semi-infinite optimization problems. Beyond these classics, recent or even brand-new results are also discussed. Perturbations in the data set and their impact on the solutions are inves- For example, Galvin’s charming list-colouring theorem (5.4.4) has appeared tigated in chapter (10), Stability and Well-Posedness, with an analysis of in 1995, or Thomassen’s pearl on 5-choosability of planar graphs in 1994. the optimal value function and the optimal-set mapping. An even more recent, very difficult result (Theoerem 8.1.1), due to Komlós Part IV, Methods of Linear Semi-Infinite Programming, gives in the and Szemerédi and to Bollobás and Thomason, state that every graph with two chapters (11) Local Reduction and Discretization Methods and (12) average degree at least cr2 contains K r as a topological minor. The paper of Simplex-Like and Exchange Methods an overview of numerical methods Komlós and Szemerédi appeared in 1996 while the work of Bollobás and for solving linear semi-infinite optimization problems. “They are described Szemerédi has appeared only this year. Isn’t it nice that a proof (of more in a conceptual form... but omitting a detailed discussion of the numerical than five pages) is already accessible in Diestel’s book? difficulties encountered in the auxiliary problems.” A grand undertaking of the past 15 years has been the proof of the so- Each chapter contains a lot of historical and bibliographical notes and called minor theorem (what a contradiction between name and signifi- hints, and ends with a collection of exercises (without solutions). These cance!) It reads: in every infinite set of finite graphs there are two such that one exercises include routine tasks and applications as well as theoretical com- is the minor of the other. The supporting theory of tree-decompositions, plements. well-quasi-orderings, minors has mainly been developed by N. Robertson For convenience, some basic concepts and properties of convex sets and and P. Seymour. The complete proof of the minor theorem is over 500 convex functions are collected in an Appendix (without proofs). So the text pages, so not surprisingly this is excluded here, but Chapter 12 on Minors, is nearly self-contained. Trees and WQO offers an excellent overview of the framework of the proof The book gives a good view of the topic. It is addressed to (graduate) stu- along with its far-reaching consequences. dents in mathematics and to scientists who are interested in the ideas The book includes 12 chapters: (1) The Basics, (2) Matching, (3) behind the theory of linear optimization. The reader is assumed to be C o n n e c t i v i t y, (4) Planar Graphs, (5) Colouring, (6) Fl ows, (7) familiar with linear algebra and elementary calculus; he should have a cer- Substructures in Dense Graphs, (8) Substructures in Sparse Graphs, (9) tain knowledge in linear programming and also in elementary topology. Ramsey Theory for Graphs, (10) Hamiltonian Cycles, (11) Random The text is carefully written, the exposition is clear and goes quite deeply Graphs, and (12) Minors, Trees, and WQU. into details. The book is more to provide a profound discussion of the sub- As far as the basic approach of the book is concerned, I just cannot state ject than to get a first insight into the topic. It may be also used as a basis it any better than the back-page review of the book does: “Viewed as a and a guideline for lectures on this subject; the authors give some propos- branch of pure mathematics, the theory of finite graphs is developed as a als of how to arrange the material for several courses. coherent subject in its own right, with its own unifying questions and As a minor complaint it should be mentioned that the list of Symbols methods. The book thus seeks to complement, not replace, the existing and Abbreviations (p. 321) unfortunately does not contain the place (page) more algorithmic treatments of the subject.” where certain notation occurs or is defined, so the reader sometimes has The book is written in a thorough and clear style. The author puts spe- trouble finding it. cial emphasis on explaining the underlying ideas, and the technical details All in all, the book leaves a remarkable impression of the concepts, tools are made as painless as possible. Some typographic novelties are introduced: and techniques in linear semi-infinite optimization. Students as well as pro- for example, there are little reminders on the margins to notations, defini- fessionals will profitably read and use it. tions, references. These may be helpful in following the text more easily; the –FRIEDRICH JUHNKE, MAGDEBURG 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. ANDRÁS FRANK
New Optimization Book Series ur MPS Council and the SIAM between that of a journal and a commercial J.E. Dennis, Jr. Council and Board of Trustees have book publisher. The journal review process approved a new joint book series, the has a certain “gatekeeper” aspect to it, appro- MPS/SIAM Series on Optimization. priate because of the significance that aca- OBoth professional societies will be demic committees attach to journal publica- responsible jointly for the scientific content tion records. Our referee process will also be of the series for which SIAM will act as pub- judgmental, but we intend to be more con- lisher. The series was announced at the May sultative with authors than is common for SIAM Optimization Meeting, and already journal editors. We will encourage and work the first manuscripts submitted to the series with the authors if we feel that the manu- are in review. Our first volume should appear script belongs in the series. Because of our early in 2000. SIAM has agreed that MPS high standards, we hope that publication in members will receive the SIAM member dis- our series will mean more to an author’s vita count on all SIAM books, not just the books than books generally mean. in this series. The advantages of SIAM as a scientific We are in the process of naming the edito- partner are self-evident. However, I feel rial board, which will be organized in the strongly that the case for SIAM as publisher standard MPS format. The first appointment is as compelling. SIAM publishes a wide vari- other than my own as Editor-in-Chief, was ety of books, and the quality of books they the appointment of Steve Wright as Co- publish has been always at least as high as the Editor for Continuous Optimization. I con- quality of commercial publishers, and now it sidered Steve’s agreement to undertake this seems generally higher. SIAM markets aggres- new task a key issue for the success of the sively at meetings, online, and by mail. At series. He is responsible, hard-working, and the same time, books published with SIAM he has written a fine monograph on interior stay in print long after a commercial publish- point methods. Thus, he understands the er would have allowed them to go out of huge difference between writing a paper and print. The prices are consistently lower than writing a successful book. We are in the for commercial publishers, though the royal- process of naming a Co-Editor for Discrete ties are competitive. SIAM will maintain a Optimization, and then we will make collec- web site for misprints and comments on each tive decisions concerning the Associate book in the series. I believe that this service Editors. I call the new series MPC, and I will be especially valuable for textbooks think of it as the logical next step after MPA because instructors can share useful software and MPB. I hope you will agree, and help to and information about exercises. make the series a success by sending us your Of course, it would have been possible to manuscripts and acting as a reviewer on collaborate with a commercial publisher in request. this venture; however, frankly, I doubt the Our aim will be to publish three to five future of commercial publishers for scientific high quality books each year. The series will books. At the Atlanta meeting next year, cover the entire spectrum of optimization. compare the per page cost at the various We welcome research monographs, textbooks booths and you will wonder, as I do, if com- at all levels, books on applications, and tuto- mercial scientific publishers have a future rials. Because our goal is to complement except for elementary texts. Let me be clear MPA and MPB, we do not intend to publish that we hope to sell a lot of books, but we proceedings or collections of papers. plan to publish fine books at low prices and The requirements for the series are simple: to keep them in print. The point here is that a manuscript must advance the understand- a commercial publisher is accountable to ing and practice of optimization, and it must investors, while SIAM is accountable to its be clearly written appropriately to its level. members, people like us. The content must be of high scientific quali- ty. Our review process will be somewhere O P T I M A 6 2 JUNE 19 9 9 18
The deadline for the next issue of OPTIMA is September15, 1999.
For the electronic version of OPTIMA, please see: http://www.ise.ufl.edu/~optima/
Application for Membersh i p
I wish to enroll as a member of the Society. Mail to: My subscription is for my personal use and not for the benefit of any library or institution. Mathematical Programming Society 3600 University City Sciences Center I will pay my membership dues on receipt of your invoice. Philadelphia PA 19104-2688 USA I wish to pay by credit card (Master/Euro or Visa). Cheques or money orders should be made payable to The Mathematical Programming
CREDITCARD NO. EXPIRY DATE Society, Inc. Dues for 1999, including sub- scription to the journal Mat h e m a t i c a l FAMILY NAME Prog r a m m i n g , are US $75.
MAILING ADDRESS Student applications: Dues are one-half the above rate. Have a faculty member verify your student status and send application with dues
TELEPHONE NO. TELEFAX NO. to above address.
EMAIL Faculty verifying status
SIGNATURE
Institution Springer ad (same as before) MATHEMATICAL PROGRAMMING SOCIETY
UN I V E R S I T Y OF FLOR I DA
Center for Applied Optimization 371 Weil PO Box 116595 FIRST CLASS MAIL Gainesville FL 32611-6595 USA
EDITOR: AREA EDITOR, DISCRETE OPTIMIZATION: BOOK REVIEW EDITOR: Karen Aardal Sebastian Ceria Robert Weismantel Department of Computer Science 417 Uris Hall Universität Magdeburg Utrecht University Graduate School of Business Fakultät für Mathematik PO Box 80089 Columbia University Universitätsplatz 2 3508 TB Utrecht New York, NY 10027-7004 D-39106 Magdeburg The Netherlands USA Germany e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] URL: http://www.cs.ruu.nl/staff/aardal.html URL: http://www.columbia.edu/~sc244/
Donald W. Hearn, FOUNDING EDITOR Elsa Drake, DESIGNER PUBLISHED BY THE MATHEMATICAL PROGRAMMING SOCIETY &
GATOREngineering® PUBLICATION SERVICES University of Florida
Jou r nal contents are subject to change by the publisher.