Oriented matroid
Introduction Oriented-matroid theory allows a combinatorial approach to the max-flow min-cut theorem. A network with the value of flow equal to the capacity of an s-t cutAn '''oriented matroid''' is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields. In comparison, an ordinary (i.e., non-oriented) matroid abstracts the dependence properties that are common both to graphs, which are not necessarily ''directed'', and to arrangements of vectors over fields, which are not necessarily ''ordered''.All oriented matroids have an underlying matroid. Thus, results on ordinary matroids can be applied to oriented matroids. However, the converse is false; some matroids cannot become an oriented matroid by ''orienting'' an underlying structure (e.g., circuits or independent sets).The distinction between matroids and oriented matroids is discussed further below.Matroids are often useful in areas such as dimension theory and algorithms.Because of an oriented matroid's inclusion of additional details about the ''oriented'' nature of a structure,its usefulness extends further into several areas including geometry and optimization.
History The first appearance of oriented matroids was in a 1966 article by George J. Minty and was confined to regular matroids.Subsequently R.T. Rockefellar (1969) suggested the problem of generalizing Minty's concept to real vector spaces. His proposal helped lead to the development of the general theory.
Background In order to abstract the concept of orientation on the edges of a graph to sets, one needs the ability to assign "direction" to the elements of a set. The way this achieved is with the following definition of ''signed sets''. * A ''signed set'', , combines a set of objects, , with an ordered ''bipartition''  of that set into two disjoint subsets:  and . : The members of  are called the ''positive elements''; members of  are the ''negative elements''. * The set  is called the ''support'' of . * The ''empty signed set'', , is defined as the empty set  combined with an (ordered) bipartition of it into two empty sets:   and . * The signed set  is the ''opposite'' of , written , if  and Given an element  of the support, we will write  for a positive element and  for a negative element. In this way, a signed set is just adding negative signs to distinguished elements. This will make sense as a "direction" only when we consider orientations of larger structures. Then the sign of each element will encode its direction relative to this orientation.
Axiomatizations Like ordinary matroids, several equivalent systems of axioms exist. (Such structures that possess multiple equivalent axiomatizations are called cryptomorphic. )=== Circuit axioms ===Let  be any set. We refer to  as the ''ground set''. Let  be a collection of ''signed sets'', each of which is ''supported'' by a subset of .If the following axioms hold for , then equivalently  is the set of ''signed circuits''for an ''oriented matroid'' on . * (C0) * (C1) (symmetric) * (C2) (incomparable) * (C3)  (weak elimination) ** ** === Vector Axioms ===The ''composition'' of signed sets  and  is the signed set  defined by  , , and . The ''vectors'' of an oriented matroid are the compositions of circuits. The vectors  of an oriented matroid satisfy the following axioms:*   * * for all , * for all , and  , there is a , such that ** ,** , and** .The ''covectors'' of an oriented matroid are the vectors of its dual oriented matroid.=== Chirotope axioms ===Let  be as above. For each non-negative integer  , a ''chirotope of rank '' is a function  that satisfies the following axioms:* (B0) ''(non-trivial)'':  is not identically zero* (B1) ''(alternating)'': For any permutation  and , , where  is the sign of the permutation. * (B2) ''(exchange)'': For any  such that  for each , then we also have .The term ''chirotope'' is derived from the mathematical notion of chirality, which is a concept abstracted from chemistry, where it is used to distinguish molecules that have the same structure except for a reflection.=== Equivalence ===Every chirotope of rank  gives rise to a set of bases of a matroid on  consisting of those -element subsets that  assigns a nonzero value. The chirotope can then sign the circuits of that matroid. If  is a circuit of the described matroid, then  where  is a basis. Then  can be signed with positive elements:and negative elements the complement. Thus a chirotope gives rise to the ''oriented bases'' of an oriented matroid. In this sense, (B0) is the nonempty axiom for bases and (B2) is the basis exchange property.
Examples Oriented matroids are often introduced (e.g., Bachem and Kern) as an abstraction for directed graphs or systems of linear inequalities. Below are the explicit constructions.=== Directed graphs ===Given a digraph, we define a signed circuit from the standard circuit of the graph by the following method. The support of the signed circuit  is the standard set of edges in a minimal cycle. We go along the cycle in the clockwise or anticlockwise direction assigning those edges whose orientation agrees with the direction to the positive elements  and those edges whose orientation disagrees with the direction to the negative elements . If  is the set of all such , then  is the set of signed circuits of an oriented matroid on the set of edges of the directed graph.A directed graphIf we consider the directed graph on the right, then we can see that there are only two circuits, namely  and . Then there are only four possible signed circuits corresponding to clockwise and anticlockwise orientations, namely ,  , , and . These four sets form the set of signed circuits of an oriented matroid on the set .=== Linear algebra ===If  is any finite subset of , then the set of minimal linearly dependent sets forms the circuit set of a matroid on . To extend this construction to oriented matroids, for each circuit  there is a minimal linear dependence:with . Then the signed circuit  has positive elements  and negative elements . The set of all such  forms the set of signed circuits of an oriented matroid on . Oriented matroids that can be realized this way are called representable.Given the same set of vectors , we can define the same oriented matroid with a chirotope . For any  let:where the right hand side of the equation is the sign of the determinant. Then  is the chirotope of the same oriented matroid on the set .=== Hyperplane arrangements ===A real hyperplane arrangement  is a finite set of hyperplanes in , each containing the origin. By picking one side of each hyperplane to be the positive side, we obtain an arrangement of half-spaces. A half-space arrangement breaks down the ambient space into a finite collection of cells, each defined by which side of each hyperplane it lands on. That is, assign each point to the signed set with   if  is on the positive side of and if  is on the negative side of . This collection of signed sets defines the set of covectors of the oriented matroid, which are the vectors of the dual oriented matroid.=== Convex polytope ===Günter M. Ziegler introduces oriented matroids via convex polytopes.
Results === Orientability ===A standard matroid is called ''orientable'' if its circuits are the supports of signed circuits of some oriented matroid. It is known that all real representable matroids are orientable. It is also known that the class of orientable matroids is closed under taking minors, however the list of forbidden minors for orientable matroids is known to be infinite. In this sense, oriented matroids is a much stricter formalization than regular matroids.=== Duality ===Just as a matroid has a unique duals, an oriented matroid has a unique dual, often called its "orthogonal dual". What this means is that the underlying matroids are dual and that the cocircuits are signed so that they are "orthogonal" to every circuit. Two signed sets are said to be ''orthogonal'' if the intersection of their supports is empty or if the restriction of their positive elements to the intersection and negative elements to the intersection form two nonidentical and non-opposite signed sets. The existence and uniqueness of the dual oriented matroid depends on the fact that every signed circuit is orthogonal to every signed cocircuit. To see why orthogonality is necessary for uniqueness one needs only to look to the digraph example above. We know that for planar graphs the dual of the circuit matroid is the circuit matroid of the graph's planar dual. Thus there are as many different dual pairs of oriented matroids based on the matroid of the graph as there are ways to orient the graph and in a corresponding way its dual.To see the explicit construction of this unique orthogonal dual oriented matroid, consider an oriented matroid's chirotope . If we consider a list of elements of  as a cyclic permutation then we define  to be the sign of the associated permutation. If  is defined as:then  is the chirotope of the unique orthogonal dual oriented matroid.=== Topological representation ===distinct from any line arrangement.Not all oriented matroids are representable—that is, not all have realizations as point configurations, or, equivalently, hyperplane arrangements. However, in some sense, all oriented matroids come close to having realizations are hyperplane arrangements. In particular, the '''Folkman–Lawrence topological representation theorem''' states that any oriented matroid has a realization as an arrangement of pseudospheres. A -dimensional ''pseudosphere'' is an embedding of  such that there exists a homeomorphism  so that  embeds  as an equator of . In this sense a pseudosphere is just a tame sphere (as opposed to wild spheres). A ''pseudosphere arrangement in '' is a collection of pseudospheres that intersect along pseudospheres. Finally, the Folkman–Lawrence topological representation theorem states that every oriented matroid of rank  can be obtained from a pseudosphere arrangement in . It is named after Jon Folkman and Jim Lawrence, who published it in 1978.=== Geometry ===Minkowski sum of the sets, which is the union of the sums having exactly one point from each summand set; for the displayed sets, the sixteen sums are distinct points, which are displayed in red: The right-hand red sum points are the sums of the left-hand red summand points. The convex hull of the sixteen red points is shaded in pink. In the pink interior of the right-hand sumset lies exactly one plus-symbol, which is the (unique) sum of the plus-symbols from the right-hand side. The right-hand plus symbol is indeed the sum of the four plus-symbols from the left-hand sets, precisely two points from the original non-convex summand sets and two points from the convex hulls of the remaining summand sets.|A zonotope, which is a Minkowski sum of line segments, is a fundamental model for oriented matroids. The sixteen dark red points (on the right) form the Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus signs (+): The right plus sign is the sum of the left plus signs.The theory of oriented matroids has influenced the development of combinatorial geometry, especially the theory of  convex polytopes, zonotopes, and configurations of vectors (equivalently, arrangements of hyperplanes). Many results—Carathéodory's theorem, Helly's theorem, Radon's theorem, the Hahn–Banach theorem, the Krein–Milman theorem, the lemma of Farkas—can be formulated using appropriate oriented matroids.=== Optimization ===In convex geometry, the simplex algorithm for linear programming is interpreted as tracing a path along the vertices of a convex polyhedron. Oriented matroid theory studies the combinatorial invariants that are revealed in the sign patterns of the matrices that appear as pivoting algorithms exchange bases.The development of an axiom system for oriented matroids was initiated by R. Tyrrell Rockafellar to describe the sign patterns of the matrices arising through the pivoting operations of Dantzig's simplex algorithm; Rockafellar was inspired by Albert W. Tucker's studies of such sign patterns in "Tucker tableaux".The theory of oriented matroids has led to breakthroughs in combinatorial optimization. In linear programming, it was the language in which Robert G. Bland formulated his pivoting rule, by which the simplex algorithm avoids cycles. Similarly, it was used by Terlaky and Zhang to prove that their  criss-cross algorithms have finite termination for linear programming problems. Similar results were made in convex quadratic programming by Todd and Terlaky. It has been applied to linear-fractional programming, quadratic-programming problems, and linear complementarity problems.Outside of combinatorial optimization, oriented matroid theory also appears in convex minimization in Rockafellar's theory of "monotropic programming" and related notions of "fortified descent". Similarly, matroid theory has influenced the development of combinatorial algorithms, particularly the greedy algorithm. More generally, a greedoid is useful for studying the finite termination of algorithms.
References
Further reading === Books ===* * * * * Evar D. Nering and Albert W. Tucker, 1993, ''Linear Programs and Related Problems'', Academic Press. (elementary)*  republished by Athena Scientific of Dimitri Bertsekas, 1998. * * === Articles ===* A. Bachem, A. Wanka, Separation theorems for oriented matroids, ''Discrete Math.'' 70 (1988) 303–310. * * * *  * * R. Tyrrell Rockafellar (1969). The elementary vectors of a subspace of , in ''Combinatorial Mathematics and its Applications'', R. C. Bose and T. A. Dowling (eds. ), Univ. of North Carolina Press, pp. 104–127. * * * * * * === On the web ===**
External links *  Komei Fukuda (ETH Zentrum, Zurich) with  publications including  ''Oriented matroid programming'' (1982 Ph.D. thesis)*  Tamás Terlaky  (Lehigh University) with  publications