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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 cut

An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields.[1] 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.[2][3]

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).[4] 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.

Contents

BackgroundEdit

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 a partition of that set into two 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 a "partition" of it into two empty sets:   and  .
  • The signed set   is the opposite of  , i.e.,  , if and only 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.


AxiomatizationsEdit

Like ordinary matroids, several equivalent systems of axioms exist. (Such structures that possess multiple equivalent axiomatizations are called cryptomorphic.)

Circuit axiomsEdit

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 AxiomsEdit

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
    •  
    •  
    •  

Chirotope axiomsEdit

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.

EquivalenceEdit

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.[5] 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.

ExamplesEdit

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 graphsEdit

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 graph

If 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 algebraEdit

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 arrangementsEdit

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.[6]

Convex polytopeEdit

Ziegler introduces oriented matroids via convex polytopes.

ResultsEdit

OrientabilityEdit

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.[7] In this sense, oriented matroids is a much stricter formalization than regular matroids.

DualityEdit

Much like matroids have unique dual, oriented matroids have unique orthogonal dual. What this means is 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.[8] To see why orthogonality is necessary for uniqueness one needs only to look to the digraph example above. We know that for planar graphs, that the dual of the circuit matroid is the circuit matroid of the graph's planar dual. Thus there are as many different oriented matroids that are dual as there are ways to orient a graph and 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.[9]

Topological representationEdit

 
This is an example of a pseudoline arrangement that is 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  .[10]

GeometryEdit

 
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 of configurations of vectors (arrangements of hyperplanes).[11] 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.[12]

OptimizationEdit

 
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". [13]

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.[14] It has been applied to linear-fractional programming,[15] quadratic-programming problems, and linear complementarity problems.[16][17][18]

Outside of combinatorial optimization, OM theory also appears in convex minimization in Rockafellar's theory of "monotropic programming" and related notions of "fortified descent".[19] Similarly, matroid theory has influenced the development of combinatorial algorithms, particularly the greedy algorithm.[20] More generally, a greedoid is useful for studying the finite termination of algorithms.

ReferencesEdit

  1. ^ Rockafellar 1969. Björner et alia, Chapters 1-3. Bokowski, Chapter 1. Ziegler, Chapter 7.
  2. ^ Björner et alia, Chapters 1-3. Bokowski, Chapters 1-4.
  3. ^ Because matroids and oriented matroids are abstractions of other mathematical abstractions, nearly all the relevant books are written for mathematical scientists rather than for the general public. For learning about oriented matroids, a good preparation is to study the textbook on linear optimization by Nering and Tucker, which is infused with oriented-matroid ideas, and then to proceed to Ziegler's lectures on polytopes.
  4. ^ Björner et alia, Chapter 7.9.
  5. ^ Björner et alia, Chapter 3.5
  6. ^ M., Björner, Anders. Las Vergnas, Michel. Sturmfels, Bernd. White, Neil. Ziegler, Gunter (2000). Oriented Matroids. Cambridge University Press. ISBN 9780511586507. OCLC 776950824.
  7. ^ Björner et alia, Chapter 7.9
  8. ^ Björner et alia, Chapter 3.4
  9. ^ Björner et alia, Chapter 3.6
  10. ^ Björner et alia, Chapter 5.2
  11. ^ Bachem and Kern, Chapters 1–2 and 4–9. Björner et alia, Chapters 1–8. Ziegler, Chapter 7–8. Bokowski, Chapters 7–10.
  12. ^ Bachem and Wanka, Chapters 1–2, 5, 7–9. Björner et alia, Chapter 8.
  13. ^ Rockafellar, R. T. (1969). "The elementary vectors of a subspace of   (1967)" (PDF). In R. C. Bose and T. A. Dowling (ed.). Combinatorial Mathematics and its Applications. The University of North Carolina Monograph Series in Probability and Statistics. Chapel Hill, North Carolina: University of North Carolina Press. pp. 104–127. MR 0278972. PDF reprint.
  14. ^ Björner et alia, Chapters 8–9. Fukuda and Terlaky. Compare Ziegler.
  15. ^ Illés, Szirmai & Terlaky (1999)
  16. ^ Fukuda & Terlaky (1997)
  17. ^ Fukuda & Terlaky (1997, p. 385)
  18. ^ Fukuda & Namiki (1994, p. 367)
  19. ^ Rockafellar 1984 and 1998.
  20. ^ Lawler. Rockafellar 1984 and 1998.


Further readingEdit

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