Consider a constraint of type $$c_1x_1+c_2x_2+\cdots+c_nx_n\leq C$$ with $x_i$ binary.
We call a cover a subset of the $n$ indices such that the sum of the corresponding coefficients is higher than the right-hand-side $C$. A cover is minimal if, by removing an index of the cover, it is not a cover anymore.
Question: what is a general procedure for finding all covers and all minimal covers?