Uniform matroid
In mathematics, a uniform matroid is a matroid in which the independent sets are exactly the sets containing at most r elements, for some fixed integer r. An alternative definition is that every permutation of the elements is a symmetry.
Definition
[edit]The uniform matroid is defined over a set of elements. A subset of the elements is independent if and only if it contains at most elements. A subset is a basis if it has exactly elements, and it is a circuit if it has exactly elements. The rank of a subset is and the rank of the matroid is .[1][2]
A matroid of rank is uniform if and only if all of its circuits have exactly elements.[3]
The matroid is called the -point line.
Duality and minors
[edit]The dual matroid of the uniform matroid is another uniform matroid . A uniform matroid is self-dual if and only if .[4]
Every minor of a uniform matroid is uniform. Restricting a uniform matroid by one element (as long as ) produces the matroid and contracting it by one element (as long as ) produces the matroid .[5]
Realization
[edit]The uniform matroid may be represented as the matroid of affinely independent subsets of points in general position in -dimensional Euclidean space, or as the matroid of linearly independent subsets of vectors in general position in an -dimensional real vector space.
Every uniform matroid may also be realized in projective spaces and vector spaces over all sufficiently large finite fields.[6] However, the field must be large enough to include enough independent vectors. For instance, the -point line can be realized only over finite fields of or more elements (because otherwise the projective line over that field would have fewer than points): is not a binary matroid, is not a ternary matroid, etc. For this reason, uniform matroids play an important role in Rota's conjecture concerning the forbidden minor characterization of the matroids that can be realized over finite fields.[7]
Algorithms
[edit]The problem of finding the minimum-weight basis of a weighted uniform matroid is well-studied in computer science as the selection problem. It may be solved in linear time.[8]
Any algorithm that tests whether a given matroid is uniform, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.[9]
Related matroids
[edit]Unless , a uniform matroid is connected: it is not the direct sum of two smaller matroids.[10] The direct sum of a family of uniform matroids (not necessarily all with the same parameters) is called a partition matroid.
Every uniform matroid is a paving matroid,[11] a transversal matroid[12] and a strict gammoid.[6]
Not every uniform matroid is graphic, and the uniform matroids provide the smallest example of a non-graphic matroid, . The uniform matroid is the graphic matroid of an -edge dipole graph, and the dual uniform matroid is the graphic matroid of its dual graph, the -edge cycle graph. is the graphic matroid of a graph with self-loops, and is the graphic matroid of an -edge forest. Other than these examples, every uniform matroid with contains as a minor and therefore is not graphic.[13]
The -point line provides an example of a Sylvester matroid, a matroid in which every line contains three or more points.[14]
See also
[edit]References
[edit]- ^ Oxley, James G. (2006), "Example 1.2.7", Matroid Theory, Oxford Graduate Texts in Mathematics, vol. 3, Oxford University Press, p. 19, ISBN 9780199202508. For the rank function, see p. 26.
- ^ Welsh, D. J. A. (2010), Matroid Theory, Courier Dover Publications, p. 10, ISBN 9780486474397.
- ^ Oxley (2006), p. 27.
- ^ Oxley (2006), pp. 77 & 111.
- ^ Oxley (2006), pp. 106–107 & 111.
- ^ a b Oxley (2006), p. 100.
- ^ Oxley (2006), pp. 202–206.
- ^ Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), "Chapter 9: Medians and Order Statistics", Introduction to Algorithms (2nd ed.), MIT Press and McGraw-Hill, pp. 183–196, ISBN 0-262-03293-7.
- ^ Jensen, Per M.; Korte, Bernhard (1982), "Complexity of matroid property algorithms", SIAM Journal on Computing, 11 (1): 184–190, doi:10.1137/0211014, MR 0646772.
- ^ Oxley (2006), p. 126.
- ^ Oxley (2006, p. 26).
- ^ Oxley (2006), pp. 48–49.
- ^ Welsh (2010), p. 30.
- ^ Welsh (2010), p. 297.