min hitting set of every base of a matroid
I meant to leave this as a comment, but I don't have the reputation to do so yet.  This question was crossposted over at Mathoverflow, where I mention that the problem is NP-complete. 

See here.  

To avoid a contradiction with Chandra Chekuri's answer, I do not believe that the LP given in his answer is integral.  To see this consider the uniform matroids $U_{k,n}$, where the bases are all $k$-subsets of a $n$-set.  Note that the vector $(1/k, 1/k, \dots, 1/k)$ is a feasible solution to the LP.  Thus, if $c$ is identically 1, then the minimum value of the LP is at most $n/k$.  On the other hand, a minimum hitting for $U_{k,n}$ has size $n-k+1$.