Approximate fixed-rank closures of covering problems

Consider a 0/1 integer program min{c T x : Ax ≥ b, x ∈ {0, 1} n } where A is nonnegative. We show that if the number of minimal covers of Ax ≥ b is polynomially bounded, then there is a polynomially large lift-and-project relaxation whose value is arbitrarily close to being at least as good as that given by the rank ≤ q cuts, for any fixed q. A special case of this result is that given by set-covering problems, or, generally, problems where the coefficients in A and b are bounded.