Oriented matroid

An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space over an ordered field (particularly for partially ordered vector spaces).^[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.

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, $X$ , combines a set of objects ${\underline {X}}$ with a partition of that set into two subsets: $X^{+}$ and $X^{-}$ .

The members of

X^{+}

are called the positive elements; members of

X^{-}

are the negative elements.

The set ${\underline {X}}=X^{+}\cup X^{-}$ is called the support of $X$ .

The empty signed set, $\emptyset$ is defined in the obvious way.

The signed set $Y$ is the opposite of $X$ , i.e., $Y=-X$ , if and only if $Y^{+}=X^{-}$ and $Y^{-}=X^{+}$

Given an an element of the support $x$ , we will write $x$ for a positive element and $-x$ for a negative element. In this way, a signed set is just adding negative signs to distinguish 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 $E$ be any set. We refer to $E$ as the ground set. Let ${\mathcal {C}}$ be a collection of signed sets, each of which is supported by a subset of $E$ . If the following axioms hold for ${\mathcal {C}}$ , then equivalently ${\mathcal {C}}$ is the set of signed circuits for an oriented matroid on $E$ .

(C0) $\emptyset \notin {\mathcal {C}}$

(C1) (symmetric) ${\mathcal {C}}=-{\mathcal {C}}$

(C2) (incomparable) ${\mbox{ for all }}X,Y\in {\mathcal {C}}{\mbox{ if }}{\underline {X}}\subseteq {\underline {Y}},{\mbox{ then }}X=Y{\mbox{ or }}X=-Y$

(C3) (weak elimination) ${\mbox{ for all }}X,Y\in {\mathcal {C}},X\neq -Y,{\mbox{ and }}e\in X^{+}\cap Y^{-}{\mbox{ there is a }}Z\in {\mathcal {C}}{\mbox{ such that }}$ ${\mbox{ for all }}X,Y\in {\mathcal {C}},X\neq -Y,{\mbox{ and }}e\in X^{+}\cap Y^{-}{\mbox{ there is a }}Z\in {\mathcal {C}}{\mbox{ such that }}$
- $Z^{+}\subseteq (X^{+}\cup Y^{+})\setminus \{e\}{\mbox{ and }}$
- $Z^{-}\subseteq (X^{-}\cup Y^{-})\setminus \{e\}.$

Examples

Oriented matroids are often introduced (e.g., Bachem and Kern) as an abstraction for directed graphs or systems of linear inequalities. Thus, it may be helpful to first be knowledgeable of the structures for which oriented matroids are an abstraction. Several of these topics are listed below.

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 $\textstyle {\underline {X}}$ 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 $\textstyle X^{+}$ and those edges whose orientation disagrees with the direction to the negative elements $\textstyle X^{-}$ . If $\textstyle {\mathcal {C}}$ is the set of all such $\textstyle X$ , then $\textstyle {\mathcal {C}}$ is the set of signed circuits of an oriented matroid on the set of edges of the directed graph.

If we consider the directed graph on the right, then we can see that there are only two circuits, namely $\textstyle \{(1,2),(1,3),(3,2)\}$ and $\textstyle \{(3,4),(4,3)\}$ . Then there are only four possible signed circuits corresponding to clockwise and anticlockwise orientations, namely $\textstyle \{(1,2),-(1,3),-(3,2)\}$ , $\textstyle \{-(1,2),(1,3),(3,2)\}$ , $\textstyle \{(3,4),(4,3)\}$ , and $\textstyle \{-(3,4),-(4,3)\}$ . These four sets form the set of signed circuits of an oriented matroid on the set $\textstyle \{(1,2),(1,3),(3,2),(3,4),(4,3)\}$ .

Linear algebra

If $\textstyle E$ is any finite subset of $\textstyle \mathbb {R} ^{n}$ , then the set of minimal linearly dependent sets forms the circuit set of a matroid on $\textstyle E$ . To extend this construction to oriented matroids, for each circuit $\textstyle \{v_{1},\dots ,v_{m}\}$ there is a minimal linear dependence

\sum _{i=1}^{m}\lambda _{i}v_{i}=0

with $\textstyle \lambda _{i}\in \mathbb {R}$ . Then the signed circuit $\textstyle X=\{X^{+},X^{-}\}$ has positive elements $\textstyle X^{+}=\{v_{i}:\lambda _{i}>0\}$ and negative elements $\textstyle X^{-}=\{v_{i}:\lambda _{i}<0\}$ . The set of all such $\textstyle X$ forms the set of signed circuits of an oriented matroid on $\textstyle E$ . Oriented matroids that can be realized this way are called representable.

Linear optimization

The theory of 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 studies of such sign patterns in "Tucker tableaux".^[5] Much of the theory of oriented matroids (OMs) was developed to study the combinatorial invariants of linear-optimization, particularly those visible in the basis-exchange pivoting of the simplex algorithm.^[6]

Convex polytope

For example, Ziegler introduces oriented matroids via convex polytopes.

Results

Algebra: duality and polarity

Oriented matroids have a satisfying theory of duality.^[7]

Geometry

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).^[8] 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.^[9]

Rank 3 oriented matroids are equivalent to arrangements of pseudolines (via the Folkman–Lawrence Topological Representation Theorem).^[10]^[11]

Similarly, matroid theory is useful for developing combinatorial notions of dimension, rank, etc.

In combinatorial convexity, the notion of an antimatroid is also useful.

Optimization

The theory of oriented matroids (OM) has led to break-throughs in combinatorial optimization. In linear programming, OM theory was the language in which Robert G. Bland formulated his pivoting rule by which the simplex algorithm avoids cycles; similarly, OM theory was used by Terlakey 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.^[12] The criss-cross algorithm is often studied using the theory of oriented matroids (OMs), which is a combinatorial abstraction of linear-optimization theory.^[6]^[13]

Historically, an OM algorithm for quadratic-programming problems and linear-complementarity problems was published by Michael J. Todd, before Terlaky and Wang published their criss-cross algorithms.^[6]^[14] However, Todd's pivoting-rule cycles on nonrealizable oriented matroids (and only on nonrealizable oriented matroids). Such cycling does not stump the OM criss-cross algorithms of Terlaky and Wang, however.^[6] There are oriented-matroid variants of the criss-cross algorithm for linear programming, for quadratic programming, and for the linear-complementarity problem.^[6]^[6]^[15]^[15] Oriented matroid theory is used in many areas of optimization, besides linear programming. OM theory has been applied to linear-fractional programming^[16] quadratic-programming problems, and linear complementarity problems.^[15]^[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.

References

^ Rockafellar 1969. Björner et alia, Chapters 1-3. Bokowski, Chapter 1. Ziegler, Chapter 7.
^ Björner et alia, Chapters 1-3. Bokowski, Chapters 1-4.
^ 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.
^ Björner et alia, Chapter 7.9.
^ (Rockafellar 1969) harv error: multiple targets (2×): CITEREFRockafellar1969 (help):
Rockafellar, R. T. (1969). "The elementary vectors of a subspace of $R^{N}$ (1967)". In R. C. Bose and T. A. Dowling (ed.). Combinatorial Mathematics and its Applications (PDF). 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. {{cite book}}: Invalid |ref=harv (help)
^ ^a ^b ^c ^d ^e ^f Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999). "10 Linear programming". Oriented Matroids. Cambridge University Press. pp. 417–479. doi:10.1017/CBO9780511586507. ISBN 978-0-521-77750-6. MR 1744046.
^ In oriented-matroid theory, duality differs from polarity; see Bachem and Kern, Chapters 5.11, 6, 7.2.
^ Bachem and Kern, Chapters 1–2 and 4–9. Björner et alia, Chapters 1–8. Ziegler, Chapter 7–8. Bokowski, Chapters 7–10.
^ Bachem and Wanka, Chapters 1–2, 5, 7–9. Björner et alia, Chapter 8.
^ Björner et alia, Chapter 6.
^ Jon Folkman; Jim Lawrence (1978). "Oriented matroids". Journal of Combinatorial Theory, Series B. 25 (2): 199–236. doi:10.1016/0095-8956(78)90039-4.
^ Björner et alia, Chapters 8-9. Fukuda and Terlaky. Compare Ziegler.
^ The theory of 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 studies of such sign patterns in "Tucker tableaux". (Rockafellar 1969) harv error: multiple targets (2×): CITEREFRockafellar1969 (help):
Rockafellar, R. T. (1969). "The elementary vectors of a subspace of $R^{N}$ (1967)". In R. C. Bose and T. A. Dowling (ed.). Combinatorial Mathematics and its Applications (PDF). 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. {{cite book}}: Invalid |ref=harv (help)
^ Todd, Michael J. (1985). "Linear and quadratic programming in oriented matroids". Journal of Combinatorial Theory. Series B. 39 (2): 105–133. doi:10.1016/0095-8956(85)90042-5. MR 0811116. {{cite journal}}: Invalid |ref=harv (help)
^ ^a ^b ^c Fukuda & Terlaky (1997)
^ Illés, Szirmai & Terlaky (1999)
^ Fukuda & Terlaky (1997, p. 385)
^ Fukuda & Namiki (1994, p. 367)
^ Rockafellar 1984 and 1998.
^ Lawler. Rockafellar 1984 and 1998.

External links

[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.

[Rockafellar-5] (Rockafellar 1969) harv error: multiple targets (2×): CITEREFRockafellar1969 (help):
Rockafellar, R. T. (1969). "The elementary vectors of a subspace of $R^{N}$ (1967)". In R. C. Bose and T. A. Dowling (ed.). Combinatorial Mathematics and its Applications (PDF). 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. {{cite book}}: Invalid |ref=harv (help)

[OMBook-6] ^ ^a ^b ^c ^d ^e ^f Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999). "10 Linear programming". Oriented Matroids. Cambridge University Press. pp. 417–479. doi:10.1017/CBO9780511586507. ISBN 978-0-521-77750-6. MR 1744046.

[7] In oriented-matroid theory, duality differs from polarity; see Bachem and Kern, Chapters 5.11, 6, 7.2.

[8] Bachem and Kern, Chapters 1–2 and 4–9. Björner et alia, Chapters 1–8. Ziegler, Chapter 7–8. Bokowski, Chapters 7–10.

[9] Bachem and Wanka, Chapters 1–2, 5, 7–9. Björner et alia, Chapter 8.

[10] Björner et alia, Chapter 6.

[11] Jon Folkman; Jim Lawrence (1978). "Oriented matroids". Journal of Combinatorial Theory, Series B. 25 (2): 199–236. doi:10.1016/0095-8956(78)90039-4.

[12] Björner et alia, Chapters 8-9. Fukuda and Terlaky. Compare Ziegler.

[13] The theory of 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 studies of such sign patterns in "Tucker tableaux". (Rockafellar 1969) harv error: multiple targets (2×): CITEREFRockafellar1969 (help):
Rockafellar, R. T. (1969). "The elementary vectors of a subspace of $R^{N}$ (1967)". In R. C. Bose and T. A. Dowling (ed.). Combinatorial Mathematics and its Applications (PDF). 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. {{cite book}}: Invalid |ref=harv (help)

[Todd-14] Todd, Michael J. (1985). "Linear and quadratic programming in oriented matroids". Journal of Combinatorial Theory. Series B. 39 (2): 105–133. doi:10.1016/0095-8956(85)90042-5. MR 0811116. {{cite journal}}: Invalid |ref=harv (help)

[FukudaTerlaky-15] Fukuda & Terlaky (1997)

[LF99Hyperbolic-16] Illés, Szirmai & Terlaky (1999)

[FTNamiki-17] Fukuda & Terlaky (1997, p. 385)

[FukudaNamiki-18] Fukuda & Namiki (1994, p. 367)

[19] Rockafellar 1984 and 1998.

[20] Lawler. Rockafellar 1984 and 1998.

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