Uniform matroid

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In mathematics, a uniform matroid is a matroid in which 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]

  1. ^ Oxley, James G. (2006), "Example 1.2.7", Matroid Theory, Oxford Graduate Texts in Mathematics, 3, Oxford University Press, p. 19, ISBN 9780199202508 . For the rank function, see p. 26.
  2. ^ Welsh, D. J. A. (2010), Matroid Theory, Courier Dover Publications, p. 10, ISBN 9780486474397 .
  3. ^ Oxley (2006), p. 27.
  4. ^ Oxley (2006), pp. 77 & 111.
  5. ^ Oxley (2006), pp. 106–107 & 111.
  6. ^ a b Oxley (2006), p. 100.
  7. ^ Oxley (2006), pp. 202–206.
  8. ^ 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 .
  9. ^ Jensen, Per M.; Korte, Bernhard (1982), "Complexity of matroid property algorithms", SIAM Journal on Computing, 11 (1): 184–190, doi:10.1137/0211014, MR 646772 .
  10. ^ Oxley (2006), p. 126.
  11. ^ Oxley (2006, p. 26).
  12. ^ Oxley (2006), pp. 48–49.
  13. ^ Welsh (2010), p. 30.
  14. ^ Welsh (2010), p. 297.