NP-hardness even with perturbations
If you allow arbitrary $e_i$, you are in muddy ground. Are your numbers real? No way to express them finitely, NP-whatever right out the window. Rational? The input size can blow up almost arbitrarily, again nothing relevant to prove.

Note that (for very good reasons) NP-hard and its ilk talk about the worst case complexity of the problem (that is well-defined, opposed to say avergage complexity; best-case essentially tells nothing in most interesting cases). That some proof gives just one particular hard case for particular sizes, while another gives a bevy of them, is irrelevant.