Uniform matroid

In mathematics, a uniform matroid is a matroid in which every permutation of the elements is a symmetry.

Definition

The uniform matroid $U{}^r_n$ is defined over a set of $n$ elements. A subset of the elements is independent if and only if it contains at most $r$ elements. A subset is a basis if it has exactly $r$ elements, and it is a circuit if it has exactly $r+1$ elements. The rank of a subset $S$ is $\min(|S|,r)$ and the rank of the matroid is $r$ .^[1]^[2]

A matroid of rank $r$ is uniform if and only if all of its circuits have exactly $r+1$ elements.^[3]

The matroid $U{}^2_n$ is called the $n$ -point line.

Duality and minors

The dual matroid of the uniform matroid $U{}^r_n$ is another uniform matroid $U{}^{n-r}_n$ . A uniform matroid is self-dual if and only if $r=n/2$ .^[4]

Every minor of a uniform matroid is uniform. Restricting a uniform matroid $U{}^r_n$ by one element (as long as $r < n$ ) produces the matroid $U{}^r_{n-1}$ and contracting it by one element (as long as $r > 0$ ) produces the matroid $U{}^{r-1}_{n-1}$ .^[5]

Realization

The uniform matroid $U{}^r_n$ may be represented as the matroid of affinely independent subsets of $n$ points in general position in $r$ -dimensional Euclidean space, or as the matroid of linearly independent subsets of $n$ vectors in general position in an $(r+1)$ -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 $n$ -point line $U{}^2_n$ can be realized only over finite fields of $n-1$ or more elements (because otherwise the projective line over that field would have fewer than $n$ points): $U{}^2_4$ is not a binary matroid, $U{}^2_5$ 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

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

Unless $r\in\{0,n\}$ , a uniform matroid $U{}^r_n$ 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, $U{}^2_4$ . The uniform matroid $U{}^1_n$ is the graphic matroid of an $n$ -edge dipole graph, and the dual uniform matroid $U{}^{n-1}_n$ is the graphic matroid of its dual graph, the $n$ -edge cycle graph. $U{}^0_n$ is the graphic matroid of a graph with $n$ self-loops, and $U{}^n_n$ is the graphic matroid of an $n$ -edge forest. Other than these examples, every uniform matroid $U{}^r_n$ with $1 < r < n-1$ contains $U{}^2_4$ as a minor and therefore is not graphic.^[13]

The $n$ -point line provides an example of a Sylvester matroid, a matroid in which every line contains three or more points.^[14]

References

↑ 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.
↑ 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.
↑ 6.0 6.1 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 646772 .
↑ Oxley (2006), p. 126.
↑ Oxley (2006, p. 26).
↑ Oxley (2006), pp. 48–49.
↑ Welsh (2010), p. 30.
↑ Welsh (2010), p. 297.