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

Lower bounds on the lattice-free rank for packing and covering integer programs