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).

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.

The problem is hard (not in P) in general. Note that a 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).

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).