  In this paper, we present lower bounds on the rank of the split closure, the
multi-branch closure and the lattice-free closure for packing sets as a
function of the integrality gap. We also provide a similar lower bound on the
split rank of covering polyhedra. These results indicate that whenever the
integrality gap is high, these classes of cutting planes must necessarily be
applied for many rounds in order to obtain the integer hull.
