Before revision:  It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
After revision:  It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(1/w_{\min}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.