I am wondering whether there is an efficient algorithm to compute the basis of the set of vertices of a polytope.
Formally,
INPUT: a polytope $$\Xi=\{(\vec{a}_1\vec{x}+\vec{b}_1, \cdots, \vec{a}_m\vec{x}+\vec{b}_m)\mid C\vec{x}\leq d\}$$ and a subspace $span(E)$ where $E=\{e_1, \cdots, e_{\ell}\}$ is a given set of vectors
OUTPUT: a basis of the linear subspace spanned by $$V(\Xi)\setminus span(E),$$ where $V(\Xi)$ denotes the set of vertices of $\Xi$.
(Note that here $\Xi$ is given as an affine mapping of a polytope, which might complicates the problem a little bit.)
One can solve the problem in a straightforward approach, but I am asking for an ideally polynomial-time algorithm, or any evidence that this is not possible (e.g., NP-hardness).