Basis of a matroid

Summary

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 edit

As an example, consider the matroid over the ground-set R2 (the vectors in the two-dimensional Euclidean plane), with the following independent sets:

{ {}, {(0,1)}, {(2,0)}, {(0,1),(2,0)}, {(0,3)}, {(0,3),(2,0)} }.

It 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:[1]

  • 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 kc elements from each category c, the bases are all sets which contain exactly kc elements from category c.
  • In a free matroid, where all subsets of the ground-set E are independent, the unique basis is E.

Properties edit

Exchange edit

All matroids satisfy the following properties, for any two distinct bases   and  :[2][3]

  • Basis-exchange property: if  , then there exists an element   such that   is a basis.
  • Symmetric basis-exchange property: if  , then there exists an element   such that both   and   are bases. Brualdi[4] showed that it is in fact equivalent to the basis-exchange property.
  • Multiple symmetric basis-exchange property: if  , then there exists a subset   such that both   and   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   from   to  , such that for every  ,   is a basis. Brualdi[4] showed that it is equivalent to the basis-exchange property.
  • Partition basis-exchange property: For each partition   of   into m parts, there exists a partition   of   into m parts, such that for every  ,   is a basis.[5]

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   need not be unique. Regular matroids have the unique exchange property, meaning that for some  , the corresponding b is unique.[6]

Cardinality edit

It follows from the basis exchange property that no member of   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 edit

It is conjectured that all matroids satisfy the following property:[2] 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 edit

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   to be a pair  , where   is the ground-set and   is a collection of subsets of  , called "bases", with the following properties:[7][8]

(B1) There is at least one base --   is nonempty;
(B2) If   and   are distinct bases, and  , then there exists an element   such that   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 edit

If   is a finite matroid, we can define the orthogonal or dual matroid   by calling a set a basis in   if and only if its complement is in  . It can be verified that   is indeed a matroid. The definition immediately implies that the dual of   is  .[9]: 32 [10]

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

(B2*) If   and   are distinct bases, and  , then there exists an element   such that   is a basis.

Circuits edit

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   to be a pair  , where   is the ground-set and   is a collection of subsets of  , called "circuits", with the following properties:[8]

(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   contains a circuit.

Another property of circuits is that, if a set   is independent, and the set   is dependent (i.e., adding the element   makes it dependent), then   contains a unique circuit  , and it contains  . This circuit is called the fundamental circuit of   w.r.t.  . It is analogous to the linear algebra fact, that if adding a vector   to an independent vector set   makes it dependent, then there is a unique linear combination of elements of   that equals  .[10]

See also edit

  • 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.

References edit

  1. ^ Ardila, Federico (2007). "Matroids, lecture 3". youtube. Archived from the original on 2020-02-14.
  2. ^ a b Bonin, Joseph E.; Savitsky, Thomas J. (2016-01-01). "An infinite family of excluded minors for strong base-orderability". Linear Algebra and Its Applications. 488: 396–429. arXiv:1507.05521. doi:10.1016/j.laa.2015.09.055. ISSN 0024-3795. S2CID 119161534.
    • Joseph E. Bonin; Thomas J. Savitsky (April 2016). "Excluded Minors for (Strongly) Base-Orderable Matroids" (PDF).
  3. ^ "Matroids Lecture 2: Bases". YouTube.
  4. ^ a b Brualdi, Richard A. (1969-08-01). "Comments on bases in dependence structures". Bulletin of the Australian Mathematical Society. 1 (2): 161–167. doi:10.1017/S000497270004140X. ISSN 1755-1633.
  5. ^ Greene, Curtis; Magnanti, Thomas L. (1975-11-01). "Some Abstract Pivot Algorithms". SIAM Journal on Applied Mathematics. 29 (3): 530–539. doi:10.1137/0129045. hdl:1721.1/5113. ISSN 0036-1399.
  6. ^ McGuinness, Sean (2014-07-01). "A base exchange property for regular matroids". Journal of Combinatorial Theory, Series B. 107: 42–77. doi:10.1016/j.jctb.2014.02.004. ISSN 0095-8956.
  7. ^ Welsh, D. J. A. (1976), Matroid Theory, L.M.S. Monographs, vol. 8, Academic Press, ISBN 978-0-12-744050-7, Zbl 0343.05002. Section 1.2, "Axiom Systems for a Matroid", pp. 7–9.
  8. ^ a b Federico, Ardila (2012). "Matroids: Lecture 6". Youtube.
  9. ^ White, Neil, ed. (1986), Theory of Matroids, Encyclopedia of Mathematics and its Applications, vol. 26, Cambridge: Cambridge University Press, ISBN 978-0-521-30937-0, Zbl 0579.00001
  10. ^ a b Ardila, Federico (2012). "Matroids lecture 7". Youtube.