M.W. Padberg, Perfect zero-one matrices, Math. Programming 6, (1974), 180-196.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 1, and for each vertex v of G, G n fvg admits a partition into p cliques of cardinality q as well as a partition into q stable sets of cardinality p. A graph is partitionable if it is a (p; q) partitionable graph for some p; q 2. Partitionable graphs were introduced by Lovasz [9] and Padberg [10] as a tool in the study of perfect graphs. A graph G is perfect if for every induced subgraph H of G, we have (H) H) Here (H) is the clique number of G, which is the cardinality of a maximum clique of G. A graph G is minimal imperfect if G is not perfect, but every induced subgraph of G is ....

....which was conjectured by Berge [2] in 1961, proved by Chudnovsky, Robertson, Seymour and Thomas [5] in 2002, says that odd cycles of length at least 5 and their complements are the only minimal imperfect graphs. Before the proof of Berge s conjecture, it was shown by Lovasz [9] and Padberg [10] that every minimal imperfect graph is a partitionable graph. Thus to prove Berge s conjecture, it suffices to show that none Preprint submitted to Elsevier Science 25 April 2003 of the partitionable graph is a counterexample. Although the final proof of Berge s conjecture given by Chudnovsky, ....

Manfred W. Padberg. Perfect zero-one matrices. Math. Programming, 6:180--196, 1974.

....1972 and states that a graph is perfect if and only if its complement is perfect. This result is known as the Perfect Graph Theorem. One of the most outstanding open problems in algorithmic graph theory is to determine the complexity of recognizing perfect graphs. Results of Lov asz [8] Padberg [10] and Bland et al. 2] imply, as it was first observed by Cameron [3] in 1982, that the problem of recognizing perfect graphs is in co NP . So far it is not known whether Humboldt Universitat zu Berlin, Institut fur Informatik, Lehrstuhl Algorithmen und Komplexitat, Unter den Linden 6, 10099 ....

M.W.Padberg, Perfect zero--one matrices, Math. Programming 6 (1974), 180--196

....theorem of Lov asz [6] Theorem 1 (The Perfect Graph Theorem) A graph G = V; E) is perfect if and only if for every induced subgraph H of G the following inequality holds: H)ff(H) jH j. This theorem was the first step toward a new approach of minimal imperfect graphs. The results of Padberg [7], Tucker [9] Chv atal [4] and Lov asz [6] provide the following list of properties satisfied by minimal imperfect graphs (we write = G) ff = ff(G) and n = jGj) for short) S1) n = ff 1; S2) for each w 2 V , G Gamma w has a unique partition into ff cliques and a unique partition ....

M. W. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974), 180--196.

....C 2k 1 , called odd antiholes, are imperfect. Clearly, each graph containing an odd hole or an odd antihole as subgraph is imperfect as well. Berge conjectured in [1] Strong Perfect Graph Conjecture) a graph is perfect i it contains neither odd holes nor odd antiholes as subgraphs. Padberg [11] introduced the notion of minimally imperfect graphs which are imperfect graphs such that all of their proper induced subgraphs are perfect. Using this term, the Strong Perfect Graph Conjecture reads: odd holes and odd antiholes are the only minimally imperfect graphs. In order to give a ....

....Conjecture reads: odd holes and odd antiholes are the only minimally imperfect graphs. In order to give a characterization of minimally imperfect graphs (and thereby to verify or falsify the Strong Perfect Graph Conjecture) many fascinating structures of such graphs have been discovered, see [8, 11]. The motivation to ask for measuring imperfectness was the following: we studied in [14, 15, 16] critical edges in perfect graphs G, i.e. edges e the deletion of which yields an imperfect graph G e. In order to decide whether the resulting graph G e is still almost perfect or already very ....

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M.W. Padberg, Perfect Zero-One Matrices, Math. Programming 6 (1974) 180-196.

....M is said to be perfect if the polyhedron P (M) fXjM:X 1 1 and X 0g has only integral extremal points. Here the vector 1 1 is a vector containing all 1 s. It follows that the problem (0) to (3) can be solved by applying a linear programming algorithm if the matrix M(I) is perfect. Padberg [18] gives a complete characterisation of perfect matrices in terms of forbidden submatrices. 3 The OCCP problem for Trees Sen et al. 21] show that the coefficient matrix M(I) associated with the integer linear program (1) to (3) is perfect if the graph G is a tree. Thus the OCCP problem for trees ....

....graph. Suppose G(I) is not perfect. Then G(I) contains a node induced p critical subgraph G 0 . It is well known that each node of a p critical subgraph G 0 is contained in exactly (G 0 ) maximal cliques of G 0 , if (G 0 ) denotes the size of a maximum clique of G 0 (Padberg [18]) As for c = 1; n the interval graph G c (I) is perfect, G 0 contains at least one bridge edge of G(I) say between G 1 (I) and G 2 (I) Thus the subgraphs G 0 G 1 (I) and G 0 G 2 (I) are not empty. Now we consider the interval p defined by f p = minff j jn j;1 2 G 0 g. ....

M.W. Padberg. Perfect Zero-One matrices. Mathematical Programming, 6 (1974) 180--196.

....antiholes. Obviously, any graph that contains an odd hole or an odd antihole is imperfect. Berge conjectured in [2] that a graph is perfect iff it contains neither odd holes nor odd antiholes as subgraphs, i.e. iff the graph is Berge (Strong Perfect Graph Conjecture, for short SPGC) Padberg [16] introduced the notion of minimally imperfect graphs; in these terms, the SPGC states that the odd holes and the odd antiholes are the only minimally imperfect graphs. Therefore, minimally imperfect Berge graphs are called monsters (since the existence of this third type of minimally imperfect ....

....in polynomial time for perfect graphs, see [8] the structure of perfect graphs is not wellunderstood. In particular, the SPGC still seems to be out of reach. On the other hand, the investigation of minimally imperfect graphs has revealed that these graphs have quite strong properties, see e.g. [5, 9, 13, 14, 15, 16, 18, 19]. That motivated us to introduce a new class of extremal cases with respect to perfectness: critically perfect graphs. We provide several examples and prove some basic properties in section 2. We are interested in relating the class CP of critically perfect graphs to well known classes of perfect ....

M.W. Padberg, Perfect Zero-One Matrices. Math. Programming 6 (1974) 180-196

....odd 6 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 = V; E) with the property that the deletion of any node results in a perfect graph, satifies the following: jV j = G) G) 1; G has exactly jV j maximum cliques, ....

M. W. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974) 180-196.

....sets in G. In order to calculate w T x over STAB(G) a description of STAB(G) as the solution set of a system of linear inequalities is required. A detailed account can be found in [3] But the linear description of the stable set polytope is known for only a few graph classes (see e.g. [2, 3, 5]) most notably for perfect graphs. Obtaining a complete list of essential, i.e. facet inducing inequalities of STAB(G) for all imperfect graphs G, seems to be a hopeless task. The present paper contributes to extending the knowledge of facet inducing inequalities to stable set polytopes ....

....their complements C n antiholes. Obviously, any graph that contains an odd hole or an odd antihole as subgraph is imperfect. Berge conjectured in [1] that a graph is perfect iff it neither contains odd holes nor odd antiholes as subgraphs (Strong Perfect Graph Conjecture, for short SPGC) Padberg [5] introduced the notion of minimally imperfect graphs, i.e. of imperfect graphs with the property that all of their proper subgraphs are perfect. In these terms, the SPGC states that the odd holes and the odd antiholes are the only minimally imperfect graphs. Therefore, minimally imperfect graphs ....

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M.W. Padberg, Perfect Zero-One Matrices. Math. Programming 6 (1974) 180-196.

....Trotter [2] a graph G is said to be partitionable if there exist two integers p and q such that G has pq 1 vertices and for every vertex v of G, the induced subgraph G n fvg admits a partition in p cliques of cardinality q and also admits a partition in q stable sets of cardinality p. Padberg [6] proved that every minimal imperfect graph is partitionable. Thus a counter example to the Strong Perfect Graph Conjecture would lie in the class of partitionable graphs. Preprint submitted to Elsevier Preprint 15 May 2000 In 1979, Chv atal, Graham, Perold and Whitesides introduced two ....

M.W. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974), 180{ 196.

....and conjectured that perfect graphs are exactly the graphs with no induced odd hole and no induced complement of an odd hole, or equivalently that minimal imperfect graphs are odd holes and their complements. This well known open conjecture is called the Strong Perfect Graph Conjecture. Padberg [10] proved that every minimal imperfect graph is partitionable. Thus a minimal imperfect graph contradicting the Strong Perfect Graph Conjecture would lie in the class of partitionable graphs. In 1979, Chvtal, Graham, Perold and Whitesides introduced two constructions for making partitionable graphs ....

M.W. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974), 180196.

....the notion of perfect graphs and conjectured that perfect graphs are exactly the graphs with no induced odd hole and no induced complement of an odd hole. This well known open conjecture is called the Strong Perfect Graph Conjecture (SPGC) Due to results of Lovsz (1972) 8] and Padberg (1974) [9], the partitionable graphs contain all minimal counter example to this conjecture. Making partitionable graphs is also of interest for the study of bus interconnexion networks, as partitionable graphs are related to directed Moore hypergraphs [2] 6] In 1979, Chvtal, Graham, Perold and Whitesides ....

M.W. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974), 180196.

....in [1] that a graph is perfect i it contains neither odd holes 1 nor odd antiholes as subgraphs; such graphs are nowadays called Berge graphs. This still open Strong Perfect Graph Conjecture has already been veri ed for several classes of Berge graphs, see next section for examples. Padberg [21] introduced the notion of minimally imperfect graphs: imperfect graphs with the property that removing any of its nodes yields a perfect graph. Using this term, the Strong Perfect Graph Conjecture reads: odd holes and odd antiholes are the only minimally imperfect graphs. In order to verify or ....

....with respect to its maximum cliques and stable sets. Theorem 2.1 (Lov asz [16] Each minimally imperfect graph G has exactly (G) G) 1 nodes and, for every node x of G, the graph G x can be partitioned into (G) cliques of size (G) and into (G) stable sets of size (G) Theorem 2. 2 (Padberg [21]) Every minimally imperfect graph G on n nodes has precisely n maximum stable sets (maximum cliques) Each node of G is contained in precisely (G) maximum stable sets ( G) maximum cliques) For every maximum clique Q (stable set S) there is a unique maximum stable set S (clique Q) with Q S = ....

M.W. Padberg, Perfect Zero-One Matrices. Math. Programming 6 (1974) 180-196

....This still open Strong Perfect Graph Conjecture has already been veri ed for several classes of F free Berge graphs, e.g. if F is a claw (Parthasarathy 1 and Ravindra [15] a diamond (Tucker [18] a clique K 4 of size 4 (Tucker [17] or a bull (Chv atal and Sbihi [4] see Figure 1. Padberg [14] introduced the notion of minimally imperfect graphs: imperfect graphs with the property that removing any of its nodes yields a perfect graph. Using this term, the Strong Perfect Graph Conjecture reads that odd holes and odd antiholes are the only minimally imperfect graphs. Therefore, minimally ....

M.W. Padberg, Perfect Zero-One Matrices. Math. Programming 6 (1974) 180-196

....holes and their complements. This conjecture is still open and is called the Strong Perfect Graph Conjecture. A weaker conjecture, stating that the complement of a perfect graph is perfect, was proposed also by Berge and proved in 1972 by Lovasz (see [11] The results of Lovasz [11] and Padberg [13] yield certain properties of minimal imperfect graphs. Following Bland, Huang and Trotter [2] given integers p, q # 2, we say that a graph G is (p, q) partitionable if, for every vertex v of G, the induced subgraph G v admits a partition into p cliques of cardinality q and also admits a ....

M.W. PADBERG, "Perfect zero-one matrices", Math. Programming 6 (1974), 180196.

....weighting with multiplicity # = #(G # ) Let us define next a partition AG = # i#I G D i , D i # D j = # for all i #= j, i, j # I G . Since G # is minimally imperfect, for every stable set S # # W there exists a maximum clique C # # W of G # for which S # # C # = # (see e.g. [26]) This implies that for every maximal stable set S # AG there exists a maximal clique i # I G , which is a maximum clique of the induced subgraph G # , i.e. for which i #W = #(G # ) and which has no common point with S in W , i.e. for which i # W # S = #. Let S # D i for such a ....

M.W. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974) 180-- 196.

....and only if i ; j 2f; 1#: #;1# 1#: # ; 1g (mod ff 1) Partitionable graphs are one of the central objects in the theory of perfect graphs due to the following theorem of Lov asz [13] Theorem 3.3 If G is a minimally imperfect graph, then G is an (ff# ) graph. Using this result, Padberg [17] proved that a minimally imperfect graph G has exactly n big cliques and n big stable sets, and J ; S T ff C is a permutation matrix. Bland et al. 4] proved that the statement remains true for (ff# ) graphs and afterwards Chv atal et al. 6] observed that the converse is also true. Theorem ....

....was originally observed in [6] As minimally imperfect graphs satisfy Condition 2) of the above theorem, Theorem 3.6 implies both Theorem 3.3 and Theorem 3.4. Next, weshowsomeknown properties of partitionable graphs. Theorem 3. 7 An (ff# ) graph G with n nodes has the following properties: 1) [17][4] G has exactly n big cliques and n big stable sets, which can be indexedas C 1 #: #C n and S 1 #: #S n , so that C i S j is empty if and only if i = j. We say that S i and C i aremates. 2) 17] 4] Every v 2 V belongs to exactly ff big stable sets and their intersection contains no other ....

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M. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974) 180-196. 12

....every node induced subgraph H , the chromatic number of H equals the number of nodes in the maximum clique of H . The connections between the integrality of the set packing polytope and the notion of a perfect graph, as defined by Berge [11,12] are given in Fulkerson [51] Lovasz [89] Padberg [97], and Chvatal [25] Theorem 4.10 Fulkerson [51] Lovasz [89] Chvatal [25] Let A be 0; 1 matrix whose columns correspond to the nodes of a graph G and whose rows are the incidence vectors of the maximal cliques of G. The graph G is perfect if and only if A is perfect. Let GA denote the ....

M.W.Padberg, Perfect zero-one matrices, Mathematical Programming 6, (1974) 180-196.

....it does not contain F k as a minor for any k 3. There are many other known minimally non ideal matrices; the reader is referred to [6] Lehman [16] and [17] shows that each nondegenerate minimally non ideal matrix is regular in a way reminiscent of the checkerboard conditions given by Padberg [20] for minimally imperfect graphs. In the following, we denote by I (respectively J) the identity matrix (respectively matrix of all 1 s) of appropriate dimension. Theorem 4 (Lehman s Theorem) Let A be a nondegenerate minimally non ideal matrix. Then r H satisfies 0 r H minf ; g and there are ....

....for x . Then for each block A i of the matrix A x , the matrix A= V Gamma S i ) has precisely jS i j rows with a minimum number of ones and these are the rows of A i . In particular, x has a unique defining matrix. 2 Covering and Packing Polyhedra A 0 1 matrix A is called perfect (see [20]) if P (A) fx 0 : A Delta x 1g is integral. Perfect matrices have been extensively studied because of their ties to the Strong Perfect Graph Conjecture and theoretical integer programming in general. It would seem that there should also be a link with ideal matrices. It is true that many of ....

M.W. Padberg, Perfect zero-one matrices, Math. Programming 6, (1974), 180-196.

....every v 2 V (G) there exists a partition of V (G) n v into stable sets of size ff and another partition into cliques of size . An immediate consequence of Lov asz characterization of perfect graphs is that every minimal imperfect graph G is partitionable with ff = ff(G) and = G) Padberg [14] has shown that if G is minimal imperfect, then G has exactly n cliques of size , every vertex of G is in exactly cliques of size , and the n Theta n incidence matrix of cliques versus vertices of G is nonsingular. Moreover the maximum cliques and the maximum stables can be numbered C 1 ; ....

M. Padberg (1974) "Perfect zero-one matrices", Math. Programming, 6, 180-196.

....dimensional polyhedron is simplicial , if it is contained in exactly n facets. A simplicial vertex has n neighbouring vertices. Neighbours share n Gamma 1 facets. If A is empty, a combinatorial coNP characterization of the integrality of P (A ; A ) is well known (Lov asz [8] Padberg [9]) If A is empty, a recent result of Lehman solves the problem (Lehman [6] Seymour [12] A common generalization of these could be a too modest goal: if for every i 2 f1; ng either the i th column of A or that of A is 0, then the nonintegrality of P (A ; A ) can be ....

....imperfect if its clique matrix is so. It is said to be partitionable, if it has n = ff 1 vertices (ff; 2 IIN) and for all v 2 V (G) G Gamma v can be partitioned both into ff cliques and into stable sets. Lov asz [8] proved that minimal imperfect graphs are partitionable and Padberg [9] proved further properties of partitionable graphs. Analogous properties have been proved for nondegenerate minimal nonideal clutters by Lehman [6] from which we extract : a pair of clutters (A; B) where B is the blocker or the antiblocker of A will be called partitionable, if they are defined ....

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M. Padberg, Perfect zero-one matrices, Math. Programming, 6 (1974) 180--196.

....for (ff; graphs. In Section 5 we use Theorem 1 to characterize fractional vertices and facets of certain types of minimally not f0; 1g polyhedra. Our characterization implies Lehman s theorem on minimally non ideal polyhedra [7] Padberg s theorem on minimally imperfect polytopes [10] and the part of Sebo [13] which correspond to nonsingular matrix equations. Matrix generalization of the singular case is going to be considered in the forthcoming paper [4] The following conventions and terminology is used throughout this article. We use lower case boldface letters to name ....

....2 V (G) here we assume that the empty graph and the clique are (ff; graphs) Lov asz [8] proved the following important theorem: Theorem 7 If G is minimal imperfect, then it is an (ff; graph. Theorem 7 provides the only known coNP characterization of perfectness, and it is used by Padberg[10] to show further properties of minimal imperfect graphs. For more on (ff; graph we refer to Chv atal et al. 1] 2 A Lemma On Matrices The following lemma contains our main argument from linear algebra. Though we need just a very special case of the lemma, we would like to state it in general ....

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M. Padberg, Perfect zero-one matrices, Math. Programming, 6 (1974) 180-- 196.

....contain the odd anti hole with seven vertices and it does not contain odd holes. In order to prove this result, we shall rely on the following properties of a minimal imperfect graph G = V; E) with n vertices: a) G has precisely n stable sets of size ff(G) and precisely n cliques of size (G) [14]: b) every vertex of G is in precisely (G) cliques of size (G) and in precisely 18. ff(G) stable sets of size ff(G) 14] c) the n stable sets can be enumerated as S 1 ; S 2 ; S n and the n cliques can be enumerated as C 1 ; C 2 ; C n in such a way that S i C j = if and ....

....on the following properties of a minimal imperfect graph G = V; E) with n vertices: a) G has precisely n stable sets of size ff(G) and precisely n cliques of size (G) 14] b) every vertex of G is in precisely (G) cliques of size (G) and in precisely 18. ff(G) stable sets of size ff(G) [14]; c) the n stable sets can be enumerated as S 1 ; S 2 ; S n and the n cliques can be enumerated as C 1 ; C 2 ; C n in such a way that S i C j = if and only if i = j [14] d) for every node v of G there exists a unique partition of G Gamma v into ff(G) cliques of size (G) ....

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M. Padberg. Perfect zero-one matrices, Math. Progr. 6 (1974) 180--196.

....property, such as transportation, assignment and minimum cost network flow problems. In resource scheduling, temporal precedence and distance constraints also constitute a totally unimodular set. Moreover, this property is sufficient but not necessary for guaranteed LP solution integrality (e.g. [12]) Applied to totally unimodular subsets of the problem constraints, LP becomes a meaningful tool for solving discrete problems. Often resource scheduling problems are discrete since real world problems have temporal granularity. For example, the aircraft utilisation problem that drove the ....

Manfred W. Padberg. Perfect zero-one matrices. Mathematical Programming, 6:180--196, 1974.

....partitionable, or (ff; graphs. Since partitionable graphs are easily seen to be imperfect, the above theorem of Lov asz provides a NP characterization of imperfectness. It also implies Lov asz s Perfect Graph Theorem stating that A graph is perfect if and only if its complement is perfect. Padberg s corollaries(1974): Let G be an (ff; graph. Let us mention four properties proved by Padberg which will be the most important for us: a. For every clique K there exists a unique ff stable set disjoint from it, let us denote it by S(K) for every ff stable set S, there exists a unique clique disjoint from ....

M. Padberg (1974) "Perfect zero-one matrices", Math. Programming, 6, 180--196.

....remarks Bland, Huang and Trotter [2] called a graph partitionable if there exist integers ff 2, 2, such that jV (G)j = ff 1 and for every v 2 V (G) there exist partitions of V (G) n v into stable sets of size ff and into cliques of size . Every minimal imperfect graph is partitionable [8]. Most known properties of minimal imperfect graphs are actually properties of all partitionable graphs. So a characterization of minimal imperfect graphs would require properties that separate those two classes of graphs. The graph in Figure 1 shows an example of a partitionable graph that admits ....

M. Padberg, Perfect zero-one matrices, Mathematical Programming 6 (1974) 180--196.

....partitioned into #(G) cliques of size #(G) and #(G) stable sets of size #(G) Moreover, for every clique C in G of size #(G) there exists a unique stable set S in G of size #(G) such that S #C = #. Finally, the inequality P v#V x v # #(G) defines a facet of STAB(G) Lovasz and Padberg [11, 17]) P3) A minimal imperfect graph G with #(G) 3 is an odd hole (Tucker [24] P4) A minimal imperfect graph G does not contain a nonempty set of nodes C such that G[V C] is disconnected and with the property that some vertex v in C is adjacent to all the remaining nodes of G (star cutset ....

M.W. Padberg, Perfect zero-one Matrices, Mathematical Programming, 6 (1974), 180--196.

....theorem of Lov asz [6] Theorem 1 (The Perfect Graph Theorem) A graph G = V; E) is perfect if and only if for every induced subgraph H of G the following inequality holds: H)ff(H) jH j. This theorem was the first step toward a new approach of minimal imperfect graphs. The results of Padberg [7], Tucker [9] Chv atal [4] and Lov asz provide the following list of properties satisfied by minimal imperfect graphs (we write = G) ff = ff(G) and n = jGj) for short) S1) n = ff 1; S2) for each w 2 V , G Gamma w has a unique partition into ff cliques and a unique partition into ....

M. W. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974), 180--196.

....odd holes and their complements. This conjecture is still open and is called the Strong Perfect Graph Conjecture. A weaker conjecture, i.e. the complement of a perfect graph is perfect, was proposed also by Berge, and proved in 1972 by Lov asz (see [11] The results of Lov asz [11] and Padberg [13] yield certain properties of minimally imperfect graphs. Following the paper by Bland, Huang and Trotter [2] for integers p; q 2, we say that a graph G is (p; q) partitionable if, for every vertex v of G, the induced subgraph G Gamma v admits a partition into p cliques of cardinality q and ....

M.W. PADBERG, "Perfect zero-one matrices", Math. Programming 6 (1974), 180-196.

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M.W. Padberg, Perfect zero-one matrices, Math. Programming 6, (1974), 180-196.

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M.W. Padberg, Perfect Zero-One Matrices. Math. Programming 6 (1974) 180--196

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M.W. Padberg, Perfect Zero-One Matrices. Math. Programming 6 (1974) 180--196

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M. W. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974), 180--196.

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M. W. Padberg, Perfect zero-one matrices, Math. Programming 6 (1974), 180--196.

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M.W. Padberg. Perfect zero-one matrices, Math. Programming 6 (1974) pp. 180-196.

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M.W. Padberg. Perfect zero-one matrices, Math. Programming 6 (1974) pp. 180-196.

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M. W. Padberg, "Perfect zero-one matrices", Mathematical Programming, 6, 180--196, 1974.

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M.W.Padberg, Perfect zero-one matrices, Math. Programming 6(1974), 180-196.

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M.W. Padberg. Perfect zero-one matrices, Math. Programming 6 (1974) pp. 180-196.

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