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22
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 17 (10 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...
Combinatorial generation of small point configurations and hyperplane arrangements
, 2003
"... A recent progress on the complete enumeration of oriented matroids enables us to generate all combinatorial types of small point configurations and hyperplane arrangements in general dimension, including degenerate ones. This extends a number of former works which concentrated on the nondegenerate ..."
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Cited by 10 (3 self)
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A recent progress on the complete enumeration of oriented matroids enables us to generate all combinatorial types of small point configurations and hyperplane arrangements in general dimension, including degenerate ones. This extends a number of former works which concentrated on the nondegenerate case and are usually limited to dimension 2 or 3. Our initial study on the complete list for small cases has shown its potential in resolving geometric conjectures. 1
Random Walks And Plane Arrangements In Three Dimensions
 AMER. MATH. MONTHLY
"... This paper explains some modern geometry and probability in the course of solving a random walk problem. Consider n planes through the origin in three dimensional Euclidean space. Assume, for simplicity, that they are in "general position". They then divide space into n(n \Gamma 1)+2 regi ..."
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Cited by 7 (1 self)
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This paper explains some modern geometry and probability in the course of solving a random walk problem. Consider n planes through the origin in three dimensional Euclidean space. Assume, for simplicity, that they are in "general position". They then divide space into n(n \Gamma 1)+2 regions. We study a random walk on these regions. Suppose the walk is in region C. Pick a pair of the planes at random. These determine a line through the origin. Pick one of the two halves of the line with equal probability. The walk now moves to the region adjacent to the chosen halfline which is closest to C. We determine the longterm stationary distribution: All regions of i sides have stationary probability proportional to i \Gamma 2. We further show that the walk is close to its stationary distribution after two steps if n is large.
Balanced Signings of Oriented Matroids and Chromatic Number
 FLOW LATTICE OF ORIENTED MATROIDS 19
, 2003
"... This paper regards an optimization problem which is an oriented matroid analogue of the graph chromatic number. There are several ways in which a `chromatic number' might be defined for more general matroids. One such formulation, introduced by Goddyn, Tarsi and Zhang [7], depends only on the s ..."
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Cited by 4 (3 self)
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This paper regards an optimization problem which is an oriented matroid analogue of the graph chromatic number. There are several ways in which a `chromatic number' might be defined for more general matroids. One such formulation, introduced by Goddyn, Tarsi and Zhang [7], depends only on the sign patterns of (signed) circuits (or cocircuits). The result is a natural invariant of an oriented matroid. The invariant can be viewed as a `discrepancy in ratio' of a pseudohyperplane arrangement, and thus should be of interest to geometers. The main theorem answers a question raised in [7]. We first state the result and some consequences, using a minimal set of definitions. Detailed definitions appear in Section 2. It is convenient to view an oriented matroid to be a matroid in which every circuit C (and cocircuit B) has been partitioned C = C C  , (and B = B subject to a standard orthogonality condition. We regard each bipartition as an unordered pair }, where one of the parts may be empty. For E(O), the reorientation I of is the new oriented matroid obtained from by repartitioning each circuit C (and cocircuit B) according to the rules } ## {C C), C  C)} and } ## {B B), B  B)}
Complete Combinatorial Generation of Small Point Configurations and Hyperplane Arrangements
 University of Waterloo
, 2001
"... A recent progress on the complete enumeration of oriented matroids enables us to generate all combinatorial types of small point configurations and hyperplane arrangements in general dimension, including degenerate ones. This extends a number of former works which concentrated on the nondegenerate c ..."
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Cited by 4 (1 self)
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A recent progress on the complete enumeration of oriented matroids enables us to generate all combinatorial types of small point configurations and hyperplane arrangements in general dimension, including degenerate ones. This extends a number of former works which concentrated on the nondegenerate case and are usually limited to dimension 2 or 3. Our initial study on the complete list for small cases has shown its potential in resolving geometric conjectures. 1
Boundary complexes of convex polytopes cannot be characterized locally
 BULL. LONDON MATH. SOC
, 1987
"... It is well known that there is no local criterion to decide the linear readability of matroids or oriented matroids. We use the setup of chirotopes or oriented matroids to derive a similar result in the context of convex polytopes. There is no local criterion to decide whether a combinatorial spher ..."
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Cited by 3 (1 self)
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It is well known that there is no local criterion to decide the linear readability of matroids or oriented matroids. We use the setup of chirotopes or oriented matroids to derive a similar result in the context of convex polytopes. There is no local criterion to decide whether a combinatorial sphere is polytopal. The proof is based on a construction technique for rigid chirotopes. These correspond, in the realizable case, to convex polytopes whose internal combinatorial structure is completely determined by its face lattice. So, a rigid chirotope is realizable over a field F if and only if its facelattice is Fpolytopal. Furthermore we prove that for every proper subfield F of the field A of real algebraic numbers there exists a 6polytope which is not realizable over F.
ShellingType Orderings of Regular CWComplexes and Acyclic Matchings of the Salvetti Complex
, 2007
"... Motivated by the work of Salvetti and Settepanella [24, Combinatorial Morse theory and minimality of hyperplane arrangements, Remark 4.5], we give a purely combinatorial description of a class of discrete Morse functions having a minimal number of critical cells for the Salvetti complex of any linea ..."
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Cited by 3 (1 self)
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Motivated by the work of Salvetti and Settepanella [24, Combinatorial Morse theory and minimality of hyperplane arrangements, Remark 4.5], we give a purely combinatorial description of a class of discrete Morse functions having a minimal number of critical cells for the Salvetti complex of any linear arrangement. We start by studying certain total orderings of the cells of shellable regular CWcomplexes, and use them to construct maximum acyclic matchings of the given complex. We apply this technique to the classical zonotope shellings. A new combinatorial stratification of the Salvetti complex allows us to paste such matchings and describe a class of maximum acyclic matchings of the whole complex. The construction can be done, so that the critical cells can be constructed from the chambers via the nbc sets. The results hold for abstract oriented matroids. 1
Balanced Signings and the Chromatic Number of Oriented Matroids
"... We consider the problem of reorienting an oriented matroid so that all its cocircuits are ‘as balanced as possible in ratio’. It is well known that any oriented matroid having no coloops has a totally cyclic reorientation, a reorientation in which every signed cocircuit B = {B +, B − } satisfies B + ..."
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Cited by 2 (2 self)
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We consider the problem of reorienting an oriented matroid so that all its cocircuits are ‘as balanced as possible in ratio’. It is well known that any oriented matroid having no coloops has a totally cyclic reorientation, a reorientation in which every signed cocircuit B = {B +, B − } satisfies B +, B − � = ∅. We show that, for some reorientation, every signed cocircuit satisfies 1/f(r) ≤ B + /B −  ≤ f(r), where f(r) ≤ 14 r 2 ln(r), and r is the rank of the oriented matroid. In geometry, this problem corresponds to bounding the discrepancies (in ratio) that occur among the Radon partitions of a dependent set of vectors. For graphs, this result corresponds to bounding the chromatic number of a connected graph by a function of its Betti number (corank) E  − V  + 1.