TITLE: A minimum set hitting every base of a matroid
QUESTION [6 upvotes]: We are given a matroid. Our goal is to find a set of elements of minimum size that has non-empty intersection with every base of the matroid. Is the problem studied before? Is it in P? For example, in a spanning tree matroid, the minimum hitting set should be a minimum cut. Thanks.
Crossposted at CSTheory.

REPLY [7 votes]: The problem is hard in general.  Note that a minimal set that intersects every base of a matroid $M$ is a dependent set in the dual matroid $M^{*}$.  Such sets are called cocircuits.  So, you are looking for the shortest cocircuit of a matroid.  
The shortest cocircuit problem (equivalently shortest circuit problem) is NP-complete in general (even for binary matroids).  See this Matroid Union post for more information on finding shortest circuits in binary matroids.