I'm a grad student in algebraic geometry, and I've encountered a problem which requires me to produce an algorithm involving matroids. Since this isn't my area of expertise, I'm hoping someone knows an algorithm which is known to do this efficiently. The problem is as follows:

I'm dealing only with matroids of rank exactly $k$ on an $n$-element set. I have a bunch of $k$-element subsets of $\{1,\ldots,n\}$. Call a matroid "good" if all of these $k$-element sets are not bases. I want to find all the matroids $M$ which are minimal among the good ones, in the sense that there is no good matroid whose independent sets are a proper subset of those of $M$.

That's the whole problem; if you care about where it came from, keep reading. I'm looking at subvarieties of the Grassmannian $G(k,n)$ which are given by ideals generated by Plücker variables. The irreducible decomposition of such a subvariety can often (though I don't think always; I'm unclear on this point still) be found by taking the subsets in the above paragraph to be the ones that appear as subscripts of the Plücker variables that generate the ideal. Since primary decomposition in Macaulay2 is slow and checking ideal equality is fast, I'm hoping to handle a lot of cases of this procedure by trying to solve it combinatorially and checking to see if I got the right answer. I'd love to talk more about what I'm doing if anyone cares.