Parameterized complexity of Hitting Set in finite VC-dimension
I think this problem is too hard. We do not know the answer to much easier problems in this family. For example, given a set of n points in the plane, and a set of (say n) unit disks, decide if there is a cover of the points by k of the unit disks. There is an easy n^O(k) time algorithm for this, and I would not be surprised if using known insights one can do n^O(sqrt{k}) (but even that is not obvious), but doing f(k)*n^{O(1)} is open, and in fact would be quite interesting. A (1+eps) approximation follows from the work of Mustafa and Ray http://portal.acm.org/citation.cfm?id=1542362.1542367.

BTW, for the continuous version where any unit disk is allowed, one can solve the problem in n^{O(k)} time. A PTAS in this case is also pretty easy using shifted grids.