Results 1  10
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35
On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, nonRadon partitions, and orientations of graphs
 TRANS. AMER. MATH. SOC
, 1983
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New finite pivoting rules for the simplex method
 Mathematics of Operations Research
, 1977
"... Mathematics of Operations Research is currently published by INFORMS. ..."
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Cited by 43 (0 self)
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Mathematics of Operations Research is currently published by INFORMS.
Generalized Kneser coloring theorems with combinatorial proofs
 INVENTIONES MATH
, 2001
"... The Kneser conjecture (1955) was proved by Lovasz (1978) using the BorsukUlam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the BorsukUlam theorem and its extensions. Only in 2000, Matousek provided the rst combinatorial proof of t ..."
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Cited by 20 (3 self)
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The Kneser conjecture (1955) was proved by Lovasz (1978) using the BorsukUlam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the BorsukUlam theorem and its extensions. Only in 2000, Matousek provided the rst combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol'nikov, AlonFranklLovasz, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver's theorem.
CrissCross Methods: A Fresh View on Pivot Algorithms
 Mathematical Programming
, 1997
"... this paper is to present mathematical ideas and ..."
Linear Complementarity and Oriented Matroids
 Journal of the Operational Research Society of Japan
, 1990
"... A combinatorial abstraction of the linear complementarity theory in the setting of oriented matroids was rst considered by M.J. Todd. In this paper, we take a fresh look at this abstraction, and attempt to give a simple treatment of the combinatorial theory of linear complementarity. We obtain new t ..."
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Cited by 12 (8 self)
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A combinatorial abstraction of the linear complementarity theory in the setting of oriented matroids was rst considered by M.J. Todd. In this paper, we take a fresh look at this abstraction, and attempt to give a simple treatment of the combinatorial theory of linear complementarity. We obtain new theorems, proofs and algorithms in oriented matroids whose specializations to the linear case are also new. For this, the notion of suciency of square matrices, introduced by Cottle, Pang and Venkateswaran, is extended to oriented matroids. Then, we prove a sort of duality theorem for oriented matroids, which roughly states: exactly one of the primal and the dual system has a complementary solution if the associated oriented matroid satisfies "weak" sufficiency. We give two different proofs for this theorem, an elementary inductive proof and an algorithmic proof using the crisscross method which solves one of the primal or dual problem by using surprisingly simple pivot rules (without any pertur...
Neighborly cubical polytopes
 Discrete & Computational Geometry
, 2000
"... Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical whose rskeleton is combinatorially equivalent to that of the convex dpolytope Cn d ndimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂Cn d of a neighborly cubical p ..."
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Cited by 11 (1 self)
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Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical whose rskeleton is combinatorially equivalent to that of the convex dpolytope Cn d ndimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂Cn d of a neighborly cubical polytope Cn d maximizes the fvector among all cubical (d − 1)spheres with 2n vertices. While we show that this is true for polytopal spheres if n ≤ d+1, we also give a counterexample for d = 4 and n = 6. Further, the existence of neighborly cubical polytopes shows that the graph of the ndimensional cube, where n ≥ 5, is “dimensionally ambiguous ” in the sense of Grünbaum. We also show that the graph of the 5cube is “strongly 4ambiguous”. In the special case d = 4, neighborly cubical polytopes have f3 = f0 4 log2 f0 4 vertices, so the facetvertex ratio f3/f0 is not bounded; this solves a problem of Kalai, Perles and Stanley studied by Jockusch.
Generation of oriented matroids – A graph theoretical approach
 Discrete Comput Geom
, 2002
"... We discuss methods for the generation of oriented matroids and of isomorphism classes of oriented matroids. Our methods are based on single element extensions and graph theoretical representations of oriented matroids, and all these methods work in general rank and for nonuniform and uniform oriente ..."
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Cited by 10 (2 self)
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We discuss methods for the generation of oriented matroids and of isomorphism classes of oriented matroids. Our methods are based on single element extensions and graph theoretical representations of oriented matroids, and all these methods work in general rank and for nonuniform and uniform oriented matroids as well. We consider two types of graphs, cocircuit graphs and tope graphs, and discuss the single element extensions in terms of localizations which can be viewed as partitions of the vertex sets of the graphs. Whereas localizations of the cocircuit graph are well characterized, there is no graph theoretical characterization known for localizations of the tope graph. In this paper we prove a connectedness property for tope graph localizations and use this for the design of algorithms for the generation of single element extensions by use of tope graphs. Furthermore we discuss similar algorithms which use the cocircuit graph. The characterization of localizations of cocircuit graphs nally leads to a backtracking algorithm which is a simple and efficient method for the generation of single element extensions. We compare this method with a recent algorithm of Bokowski and Guedes de Oliveira for uniform oriented matroids.
A Survey on Pivot Rules for Linear Programming
 ANNALS OF OPERATIONS RESEARCH. (SUBMITTED
, 1991
"... The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. Th ..."
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Cited by 9 (1 self)
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The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...
Matroidal Bijections between Graphs
, 1988
"... We study a hierarchy of five classes of bijections between the edge sets of two graphs: weak maps, strong maps, cyclic maps, orientable cyclic maps, and chromatic maps. Each of these classes contains the next one and is a natural class of mappings for some family of matroids. For example, f: E(G)t ..."
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Cited by 5 (0 self)
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We study a hierarchy of five classes of bijections between the edge sets of two graphs: weak maps, strong maps, cyclic maps, orientable cyclic maps, and chromatic maps. Each of these classes contains the next one and is a natural class of mappings for some family of matroids. For example, f: E(G)t E(H) is cyclic if every cycle (eulerian subgraph) of G is mapped onto a cycle of H. This class of mappings is natural when graphs are considered as binary matroids. A chromatic map E(G) + E(H) is induced by a (vertex) homomorphism from G to H. For such maps, the notion of a vertex is meaningful so they are natural for graphic matroids. In the same way that chromatic maps lead to the definition of X(Gtthe chromatic numberthe other classes give rise to new interesting graph parameters. For example, 4(G) is the least order of H for which there exists a cyclic bijection f: E(G)t E(H). We establish some connection between 4 and x, e.g., x(G)> i(G)> x(G)/2. The exact relation between 4 and x depends on knowledge of the chromatic number of C $ the square of the ndimensional cube. Higher powers of C, are considered, too, and tight bounds for their chromatic number are found, through