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Matroid
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In combinatorics, a branch of mathematics, a matroid or independence structure is a structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces.
There are many equivalent ways to define a matroid, and many concepts within matroid theory have a variety of equivalent formulations. Depending on the sophistication of the concept, it may be nontrivial to show that the different formulations are equivalent, a phenomenon sometimes called cryptomorphism.
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In combinatorics, a branch of mathematics, a matroid or independence structure is a structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces.
There are many equivalent ways to define a matroid, and many concepts within matroid theory have a variety of equivalent formulations. Depending on the sophistication of the concept, it may be nontrivial to show that the different formulations are equivalent, a phenomenon sometimes called cryptomorphism. Significant definitions of matroid include those in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions.
Matroid theory borrows extensively from the terminology of linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields.
Formal definitions of finite matroids
There are dozens of equivalent ways to define a (finite) matroid. Here are some of the most important. There is no one preferred or customary definition; in that respect, matroids differ from many other mathematical structures, such as groups and topologies.
Independent sets, bases, and circuits
One of the most valuable definitions is that in terms of independence. In this definition a finite matroid M is a pair (E, I), where E is a finite set and I is a collection of subsets of E (called the independent sets) with the following properties:
- The empty set is independent. (Alternatively, at least one subset of E is independent.)
- Every subset of an independent set is independent. This is sometimes called the hereditary property.
- If A and B are two independent sets and A has more elements than B, then there exists an element in A which is not in B and when added to B still gives an independent set. This is sometimes called the augmentation property or the independent set exchange property.
The first two properties are simple, and define a combinatorial structure known as an independence system, but the motivation behind the third property is not obvious. The examples in the example section below will make its motivation clearer.
A subset of E that is not independent is called dependent. A maximal independent set—that is, an independent set which becomes dependent on adding any element of E—is called a basis for the matroid. It is a basic result of matroid theory, directly analogous to a similar theorem of linear algebra, that any two bases of a matroid M have the same number of elements. This number is called the rank of M.
A circuit in a matroid M is a minimal dependent subset of E—that is, a dependent set whose proper subsets are all independent. In spite of the similarity to the definition of basis this notion has no analogue in classical linear algebra. The terminology comes from graph theory; see below.
The dependent sets, the bases, or the circuits of a matroid characterize the matroid completely. Furthermore, the collection of dependent sets, or of bases, or of circuits each has simple properties that may be taken as axioms for a matroid. For instance, one may define a matroid M to be a pair (E, B), where E is a finite set as before and B is a collection of subsets of E, called bases, with the following property:
- If A and B are distinct bases of M and a is an element of A not belonging to B, then there exists an element b belonging to B such that A - a + b is a basis. This property is sometimes called the basis exchange property.
Rank functions
If M is a matroid on E, and A is a subset of E, then a matroid on A can be defined by considering a subset of A independent if and only if it is independent in M. This allows us to talk about submatroids and about the rank of any subset of E.
The rank function r assigns a natural number to every subset of E and has the following properties:
- r(A) ≤ |A| for all subsets A of E.
- If A and B are subsets of E with A ⊆ B, then r(A) ≤ r(B).
- For any two subsets A and B of E, we have r(A ∪ B) + r(A ∩ B) ≤ r(A) + r(B).
These three properties can be used as one of the alternative definitions of a finite matroid: the independent sets are then defined as those subsets A of E with r(A) = |A|.
Closure operators
Let M be a matroid on a finite set E, defined as above. The closure cl(A) of a subset A of E is the subset of E containing A and every element x in E\A, such that there is a circuit C containing x and contained in the union of A and . This defines the closure operator, from P(E) to P(E), where P denotes the power set.
The closure operator cl from P(E) to P(E) has the following property:
- For all elements a, b of E and all subsets Y of E, if a is in cl(Y ∪ b) \ cl(Y), then b is in cl(Y ∪ a). This is sometimes called the Mac Lane–Steinitz exchange property.
In fact this may be taken as another definition of matroid – any closure operator on E with this property determines a matroid on E.
Closed sets (flats)
A set whose closure equals itself is said to be closed, or a flat of the matroid. The closed sets of a matroid are characterized by a covering partition property:
- The whole point set E is closed.
- If S and T are flats, then S n T is a flat.
- If S is a flat, then the flats T that cover S, i.e., T properly contains S but there is no flat U between S and T, partition the elements of E − S.
The class L(M) of all flats, partially ordered by set inclusion, forms a matroid lattice.
Conversely, the set A of atoms of any matroid lattice L form a matroid under the following closure operator: for a set S of atoms whose join in L is x,
- cl(S) = .
Equivalently, the flats of the matroid are the sets
- for x?L.
Thus, the lattice of flats of this matroid is naturally isomorphic to L.
Examples
Simple matroids
A matroid is called simple if it has no circuits consisting of 1 or 2 elements. That is, it has no loops and no parallel elements.
Uniform matroids
Let E be a finite set and k a natural number. One may define a matroid on E by taking every k-element subset of E to be a basis. This is known as the uniform matroid of rank k. All uniform matroids of rank at least 2 are simple.
The uniform matroid of rank 2 on n points is called the n-point line.
Discrete matroids
A matroid is called discrete if every element is a loop or a coloop. Equivalently, every proper, non-empty subset of the ground set E is a separator (loop, coloop, and separator are defined in Additional Terminology below).
Matroids from linear algebra
Matroid theory developed mainly out of a deep examination of the properties of independence and dimension in vector spaces. Matroids from vector spaces are still the main examples. There are two ways to present them.
- If E is any finite subset of a vector space V, then we can define a matroid M on E by taking the independent sets of M to be the linearly independent elements in E. We say the set E represents M. Matroids of this kind are called vector matroids.
- A matrix A with entries in a field gives rise to a matroid M on its set of columns. The dependent sets of columns in the matroid are those that are linearly dependent as vectors. This matroid is called the column matroid of A, and A is said to represent M. Column matroids are just vector matroids under another name, but there are often reasons to favor the matrix representation. (There is one technical difference: a column matroid can have distinct elements that are the same vector, but a vector matroid as defined above cannot. Usually this difference is insignificant and can be ignored, but by letting E be a multiset of vectors one brings the two definitions into complete agreement.)
A matroid that is equivalent to a vector matroid, although it may be presented differently, is called representable. If M is equivalent to a vector matroid over a field F, then we say M is representable over F. For instance, although a graphic matroid (see below) is presented in terms of a graph, it is also representable by vectors over any field. A basic problem in matroid theory is to determine whether a given matroid M is representable over a given field F. Whitney found one solution to this problem when F is a field with two elements (see "Binary matroids", below), but the general situation is famously complicated.
Matroids from graph theory
A second original source for the theory of matroids is graph theory.
Every finite graph (or multigraph) G gives rise to a matroid as follows: take as E the set of all edges in G and consider a set of edges independent if and only if it does not contain a simple cycle. Such an edge set is called a forest in graph theory. This is called the cycle matroid or graphic matroid of G ; it is usually written M(G).
Any matroid that is equivalent to the cycle matroid of a (multi)graph, even if it is not presented in terms of graphs, is called a graphic matroid. The matroids that are graphic have been characterized by Tutte.
The bicircular matroid of a graph is defined by calling a set of edges independent if every connected subset contains at most one cycle.
In any (directed) graph G let E and F be two distinguished sets of vertices. In the set F, define a subset U to be independent if there are |U| vertex-disjoint paths from E onto U. This defines a matroid on F called a Gammoid.
Matroids from biased graphs
Graphic matroids have been generalized to matroids from signed graphs, gain graphs, and biased graphs. A graph G with a distinguished linear class B of cycles, known as a "biased graph", has two matroids, known as the frame matroid and the lift matroid of the biased graph (G,B). If every cycle belongs to the distinguished class, these matroids coincide with the cycle matroid of G. If no cycle is distinguished, the frame matroid is the bicircular matroid of G.
A signed graph, whose edges are labeled by signs, and a gain graph, which is a graph whose edges are labeled orientably from a group, have the same two matroids since it gives rise to a biased graph.
Frame matroids
A matroid M is called a frame matroid if it, or a matroid that contains it, has a basis such that all the points of M are contained in the lines that join pairs of basis elements.
Transversal matroids
Given a set of "points", E, and a class A of subsets of E, a transversal of A is a subset S of E such that there is a one-to-one function f from S to A by which x belongs to f (x) for each x in S. (A may have repeated members, which are treated as separate subsets of E.) The set of transversals forms the class of independent sets of a matroid, called the transversal matroid of (E, A). These matroids are a special simple case of the Gammoid structures defined above. Simply define a bipartite graph with A on the left, E on the right, and join X in A to x in E if x is an element of X. The distinguished sets in the gammoid are A and E respectively.
Matroids from field extensions
A third original source of matroid theory is field theory.
An extension of a field gives rise to a matroid. Suppose F and K are fields with K containing F. Let E be any finite subset of K. Define a subset S of E to be independent if the extension field F[S] has transcendence degree equal to |S|.
A matroid that is equivalent to a matroid of this kind is called an algebraic matroid. The problem of characterizing algebraic matroids is extremely difficult; little is known about it.
The Fano matroid
Matroids with a small number of elements are often pictured as in the diagram. The dots are the elements of the underlying set, and a curve has been drawn through every 3-element circuit. The diagram shows a rank 3 matroid called the Fano matroid, an example that appeared in the original 1935 paper of Whitney.
The name arises from the fact that the Fano matroid is the projective plane of order 2, known as the Fano plane, whose coordinate field is the 2-element field. This means the Fano matroid is the vector matroid associated to the seven nonzero vectors in a three-dimensional vector space over a field with two elements.
It is known from projective geometry that the Fano matroid is not representable by any set of vectors in a real or complex vector space (or in any vector space over a field whose characteristic differs from 2).
A less famous example is the anti-Fano matroid, defined in the same way as the Fano matroid with the exception that the circle in the above diagram is missing. The anti-Fano matroid is representable over a field if and only if its characteristic differs from 2.
The direct sum of a Fano matroid and an anti-Fano matroid is the simplest example for a matroid which is not representable over any field.
Non-examples
On the other hand, consider this non-example: let E be a set of pairs (v,c) where v ranges over the vertices of a graph and c ranges over the set . Let the independent sets be the sets of pairs that associate only one color with each vertex and do not associate the same color with two adjacent vertices; that is, they represent valid graph colorings. The empty set is a valid three-coloring, and any subset of a valid three-coloring is a valid three-coloring, but the exchange property does not hold, because it's possible to have two maximal three-colored subgraphs of different sizes, as shown to the right. It's no surprise that this is not a matroid, since if it were, it would give us a greedy algorithm for the NP-complete 3-coloring problem, showing P = NP.
Basic constructions
Let M be a matroid with an underlying set of elements E, and let N be another matroid on underlying set F.
There are some standard ways to make new matroids out of old ones.
- Restriction. If S is a subset of E, the restriction of M to S, written M|S, is the matroid on underlying set S whose independent sets are the independent sets of M that are contained in S. Its circuits are the circuits of M that are contained in S and its rank function is that of M restricted to subsets of S. In linear algebra, this corresponds to restricting to the subspace generated by the vectors in S.
- Contraction. If T is a subset of E, the contraction of M by T, written M/T, is the matroid on the underlying set E − T whose rank function is In linear algebra, this corresponds to looking at the quotient space by the linear space generated by the vectors in T, together with the images of the vectors in E - T.
- Minors. A matroid N that is obtained from M by a sequence of restriction and contraction operations is called a minor of M. We say M contains N as a minor.
- Direct sum. The direct sum of M and N is the matroid whose underlying set is the disjoint union of E and F, and whose independent sets are the disjoint unions of an independent set of M with an independent set of N.
- Theorem. A matroid is the direct sum of its restrictions to its irreducible separators .
- Matroid union. The union of M and N is the matroid whose underlying set is the union (not the disjoint union) of E and F, and whose independent sets are those subsets whose intersections with both E and F are independent. Usually the term "union" is applied when E = F, but that assumption is not essential. If E and F are disjoint, the union is the direct sum.
Additional terminology
Let M be a matroid with an underlying set of elements E.
- A subset of E spans M if its closure is E. A set is said to span a closed set K if its closure is K.
- A maximal closed proper subset of E is called a coatom or copoint or hyperplane of M. An equivalent definition: A coatom is a subset of E that does not span M, but such that adding any other element to it does make a spanning set.
- An element that forms a single-element circuit of M is called a loop. Equivalently, an element is a loop if it belongs to no basis.
- An element that belongs to no circuit is called a coloop. Equivalently, an element is a coloop if it belongs to every basis.
- If a two-element set is a circuit of M, then f and g are parallel in M.
- A simple matroid obtained from M by deleting all loops and deleting one element from each 2-element circuit until no 2-element circuits remain is called a simplification of M.
- A separator of M is a subset S of E such that A proper separator is a separator that is neither E nor the empty set. An irreducible separator is a separator that contains no other non-empty separator. The irreducible separators partition the ground set E.
- A matroid which cannot be written as the direct sum of two nonempty matroids, or equivalently which has no proper separators, is called connected or irreducible.
- A maximal irreducible submatroid of M is called a component of M. A component is the restriction of M to an irreducible separator, and contrariwise, the restriction of M to an irreducible separator is a component.
Further topics
Regular matroids
A matroid is regular if it can be represented by a totally unimodular matrix (a matrix whose square submatrices all have determinants equal to 0, 1, or −1). Tutte proved that the following three properties of a matroid are logically equivalent:
- M is regular.
- M is representable over every field.
- M has no minor that is a four-point line or a Fano plane or its dual.
For this he used his difficult homotopy theorem. Simpler proofs have since been found.
Seymour's decomposition theorem states that all regular matroids can be built up in a simple way as the clique-sum of graphic matroids, their duals, and one special matroid. This theorem has major consequences for linear programming involving totally unimodular matrices.
Binary matroids
A matroid that is representable over the two-element field is called a binary matroid. Binary matroids include graphic and regular matroids. They have many of the nice properties of those types of matroid. Whitney and Tutte found famous characterizations. Addition of sets is symmetric difference. The following properties of a matroid M are equivalent:
- M is binary.
- In M, every sum of circuits is a union of disjoint circuits (Whitney).
- M has no minor that is a four-point line (Tutte).
The monograph of Recski compiled 8 equivalent definitions of binary matroids.
Forbidden minors
Suppose L is a list of matroids. The class Ex(L) of all matroids that do not contain as a minor any member of the list is said to be characterized by forbidden minors (or excluded minors). Some of the great theorems of matroid theory characterize natural classes of matroids by forbidden minors. Three examples due to Tutte:
- Binary matroids (see above).
- Regular matroids (see above).
- Graphic matroids are the matroids that have no minor that is the four-point line (which is self-dual), the Fano plane or its dual, the dual of the cycle matroid M(K5), or the dual of the cycle matroid M(K3,3).
It is easy to show that the matroids representable over a fixed field can be characterized by a list of forbidden minors. A famous outstanding problem (Rota's conjecture) is to prove that this list is finite for finite fields. This has been solved only for the fields of up to four elements (and the exact lists are known for those fields, but one cannot expect exact lists for larger fields). The problem is significant because there are matroid properties that can be characterized by forbidden minors but not by a finite list of them—for example, the property of being representable over the real numbers.
The Robertson-Seymour Theorem, whose full proof runs to more than 500 pages, states that every matroid property of graphic matroids characterized by a list of forbidden minors can be characterized by a finite list—in other words, if an infinite list L includes the forbidden minors for graphic matroids, then Ex(L) = Ex(L’) for some finite list L’.
Matroid duality
If M is a finite matroid, we can define the dual matroid M* by taking the same underlying set and calling a set a basis in M* if and only if its complement is a basis in M. It is not difficult to verify that M* is a matroid and that the dual of M* is M.
The dual can be described equally well in terms of other ways to define a matroid. For instance:
- A set is independent in M* if and only if its complement spans M.
- A set is a circuit of M* if and only if its complement is a coatom in M.
- The rank function of the dual is r*(S) = |S|- r(E) + r(E\S).
A main result is the matroid version of Kuratowski's theorem: The dual of a graphic matroid M is a graphic matroid if and only if M is the matroid of a planar graph.
A simpler result is that the dual of a vector matroid representable over a particular field F is also representable over F.
It is known that the dual of a transversal matroid is a strict gammoid and vice versa. See Box 15.2 in the monograph of Recski for the relations among gammoids, strict gammoids, transversal and fundamental transversal matroids
Greedy algorithms
A weight function on a finite set E is a function w: E → R+ from E to the nonnegative real numbers. Abusing notation, such a weight function w can be extended to subsets S ⊆ E by defining
- w(S) = ∑s ∈ Sw(s).
Suppose E is a finite set, F a nonempty family of subsets of E such that any subset of any element of F also belongs to F, and w: F → R+ a weight function on F. A greedy algorithm for (E, F, w) is any algorithm that attempts to construct a maximum weight element of F as follows:
- 1. Let F0 = ∅.
- 2. For i ≥ 0:
3. Let Zi = .
4. If Zi = ∅, terminate and return Fi.
5. Otherwise, choose an element y ∈ Zi such that w(y) = max, let Fi+1 = Fi ∪ and continue.
The following two theorems establish a correspondence between matroids and sets (E, F) as defined above for which greedy algorithms do give maximum weight solutions.
- Theorem 1: If E in (E, F, w) is the underlying set of a matroid M and F is the set of independent sets in M, then any greedy algorithm for (E, F, w) constructs a maximum weight element of F.
- Theorem 2: If every greedy algorithm for the pair (E, F) constructs a maximum weight element of F for every choice of weight function w: F → R+ then F is the set of independent sets of a matroid M with underlying set E.
The notion of matroid has been generalized to allow for other types of sets on which greedy algorithms give optimal solutions; see greedoid for more information.
Infinite matroids
The theory of infinite matroids is much more complicated than that of finite matroids and forms a subject of its own. One of the difficulties is that there are many reasonable and useful definitions, none of which captures all the important aspects of finite matroid theory. For instance, it seems to be hard to have bases, circuits, and duality together in one notion of infinite matroids.
The simplest definition of an infinite matroid is to require finite rank; that is, the rank of E is finite. This theory is similar to that of finite matroids except for the failure of duality due to the fact that the dual of an infinite matroid of finite rank does not have finite rank. Finite-rank matroids include any subsets of finite-dimensional vector spaces and of field extensions of finite transcendence degree.
The next simplest infinite generalization is finitary matroids. A matroid is finitary if it has the property that
Equivalently, every dependent set contains a finite dependent set.
Examples are linear dependence of arbitrary subsets of infinite-dimensional vector spaces (but not infinite dependencies as in Hilbert and Banach spaces), and algebraic dependence in arbitrary subsets of field extensions of possibly infinite transcendence degree. Again, the class of finitary matroid is not self-dual, because the dual of a finitary matroid is not finitary.
Finitary infinite matroids are studied in model theory, a branch of mathematical logic with strong ties to algebra.
Terminology
The name "matroid" was introduced by Hassler Whitney when he invented matroids. The terminology of matroid theory borrows heavily from linear algebra and graph theory.
The name combinatorial pregeometry was introduced by Gian-Carlo Rota as a replacement for "the ineffably cacophonous [sic] term 'matroid'". Rota also proposed combinatorial geometry to replace "simple matroid". However, over time, virtually everyone has abandoned the substitute names.
See also
Researchers
External links
Contains several other equivalent definitions of matroids.
- Locke, S. C. : .
- Pagano, Steven R. : .
- Kingan, Sandra: . Lots of links.
- Truemper, Klaus .
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