Английская Википедия:Basis of a matroid

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Шаблон:Short description In mathematics, a basis of a matroid is a maximal independent set of the matroid—that is, an independent set that is not contained in any other independent set.

Examples

As an example, consider the matroid over the ground-set R² (the vectors in the two-dimensional Euclidean plane), with the following independent sets: Шаблон:Block indentIt has two bases, which are the sets {(0,1),(2,0)} , {(0,3),(2,0)}. These are the only independent sets that are maximal under inclusion.

The basis has a specialized name in several specialized kinds of matroids:

• In a graphic matroid, where the independent sets are the forests, the bases are called the spanning forests of the graph.
• In a transversal matroid, where the independent sets are endpoints of matchings in a given bipartite graph, the bases are called transversals.
• In a linear matroid, where the independent sets are the linearly-independent sets of vectors in a given vector-space, the bases are just called bases of the vector space. Hence, the concept of basis of a matroid generalizes the concept of basis from linear algebra.
• In a uniform matroid, where the independent sets are all sets with cardinality at most k (for some integer k), the bases are all sets with cardinality exactly k.
• In a partition matroid, where elements are partitioned into categories and the independent sets are all sets containing at most k_c elements from each category c, the bases are all sets which contain exactly k_c elements from category c.
• In a free matroid, where all subsets of the ground-set E are independent, the unique basis is E.

Properties

Exchange

All matroids satisfy the following properties, for any two distinct bases <math>A</math> and <math>B</math>:

• Basis-exchange property: if <math>a\in A\setminus B</math>, then there exists an element <math>b\in B\setminus A</math> such that <math>(A \setminus \{ a \}) \cup \{b\}</math> is a basis.
• Symmetric basis-exchange property: if <math>a\in A\setminus B</math>, then there exists an element <math>b\in B\setminus A</math> such that both <math>(A \setminus \{ a \}) \cup \{b\}</math> and <math>(B \setminus \{ b \}) \cup \{a\}</math> are bases. Brualdi showed that it is in fact equivalent to the basis-exchange property.
• Multiple symmetric basis-exchange property: if <math>X\subseteq A\setminus B</math>, then there exists a subset <math>Y\subseteq B\setminus A</math> such that both <math>(A \setminus X) \cup Y</math> and <math>(B \setminus Y) \cup X</math> are bases. Brylawski, Greene, and Woodall, showed (independently) that it is in fact equivalent to the basis-exchange property.
• Bijective basis-exchange property: There is a bijection <math>f</math> from <math>A</math> to <math>B</math>, such that for every <math>a\in A\setminus B</math>, <math>(A \setminus \{ a \}) \cup \{f(a)\}</math> is a basis. Brualdi showed that it is equivalent to the basis-exchange property.
• Partition basis-exchange property: For each partition <math>(A_1, A_2, \ldots, A_m)</math> of <math>A</math> into m parts, there exists a partition <math>(B_1, B_2, \ldots, B_m)</math> of <math>B</math> into m parts, such that for every <math>i\in[m]</math>, <math>(A \setminus A_i) \cup B_i</math> is a basis.

However, a basis-exchange property that is both symmetric and bijective is not satisfied by all matroids: it is satisfied only by base-orderable matroids.

In general, in the symmetric basis-exchange property, the element <math>b\in B\setminus A</math> need not be unique. Regular matroids have the unique exchange property, meaning that for some <math>a\in A\setminus B</math>, the corresponding b is unique.

Cardinality

It follows from the basis exchange property that no member of <math>\mathcal{B}</math> can be a proper subset of another.

Moreover, all bases of a given matroid have the same cardinality. In a linear matroid, the cardinality of all bases is called the dimension of the vector space.

Neil White's conjecture

It is conjectured that all matroids satisfy the following property: For every integer t ≥ 1, If B and B' are two t-tuples of bases with the same multi-set union, then there is a sequence of symmetric exchanges that transforms B to B'.

Characterization

The bases of a matroid characterize the matroid completely: a set is independent if and only if it is a subset of a basis. Moreover, one may define a matroid <math>M</math> to be a pair <math>(E,\mathcal{B})</math>, where <math>E</math> is the ground-set and <math>\mathcal{B}</math> is a collection of subsets of <math>E</math>, called "bases", with the following properties:

(B1) There is at least one base -- <math>\mathcal{B}</math> is nonempty;

(B2) If <math>A</math> and <math>B</math> are distinct bases, and <math>a\in A\setminus B</math>, then there exists an element <math>b\in B\setminus A</math> such that <math>(A \setminus \{ a \}) \cup \{b\} </math> is a basis (this is the basis-exchange property).

(B2) implies that, given any two bases A and B, we can transform A into B by a sequence of exchanges of a single element. In particular, this implies that all bases must have the same cardinality.

Duality

If <math>(E,\mathcal{B})</math> is a finite matroid, we can define the orthogonal or dual matroid <math>(E,\mathcal{B}^*)</math> by calling a set a basis in <math>\mathcal{B}^* </math> if and only if its complement is in <math>\mathcal{B}</math>. It can be verified that <math>(E,\mathcal{B}^*)</math> is indeed a matroid. The definition immediately implies that the dual of <math>(E,\mathcal{B}^*)</math> is <math>(E,\mathcal{B})</math>.Шаблон:Rp

Using duality, one can prove that the property (B2) can be replaced by the following:

(B2*) If <math>A</math> and <math>B</math> are distinct bases, and <math>b\in B\setminus A</math>, then there exists an element <math>a\in A\setminus B</math> such that <math>(A \setminus \{ a \}) \cup \{b\} </math> is a basis.

Circuits

A dual notion to a basis is a circuit. A circuit in a matroid is a minimal dependent set—that is, a dependent set whose proper subsets are all independent. The terminology arises because the circuits of graphic matroids are cycles in the corresponding graphs.

One may define a matroid <math>M</math> to be a pair <math>(E,\mathcal{C})</math>, where <math>E</math> is the ground-set and <math>\mathcal{C}</math> is a collection of subsets of <math>E</math>, called "circuits", with the following properties:

(C1) The empty set is not a circuit;

(C2) A proper subset of a circuit is not a circuit;

(C3) If C₁ and C₂ are distinct circuits, and x is an element in their intersection, then <math>C_1 \cup C_2 \setminus \{x\}</math> contains a circuit.

Another property of circuits is that, if a set <math>J</math> is independent, and the set <math>J \cup \{x\}</math> is dependent (i.e., adding the element <math>x</math> makes it dependent), then <math>J \cup \{x\}</math> contains a unique circuit <math>C(x,J)</math>, and it contains <math>x</math>. This circuit is called the fundamental circuit of <math>x</math> w.r.t. <math>J</math>. It is analogous to the linear algebra fact, that if adding a vector <math>x</math> to an independent vector set <math>J</math> makes it dependent, then there is a unique linear combination of elements of <math>J</math> that equals <math>x</math>.

See also

• Matroid polytope - a polytope in Rⁿ (where n is the number of elements in the matroid), whose vertices are indicator vectors of the bases of the matroid.