On the Rank of Cutting-Plane Proof Systems

We introduce a natural abstraction of propositional proof systems that are based on cutting planes. This leads to a new class of proof systems that includesmanywell-knownmethods, such as Gomory-Chvátal cuts, lift-and-project cuts, Sherali-Adams cuts, or split cuts. The rank of a proof system corresponds to the number of rounds that is needed to show the nonexistence of integral solutions. We exhibit a family of polytopes without integral points contained in the n-dimensional 0/1-cube that has rank Ω(n/ log n) for every proof system in our class. In fact, we show that whenever some cutting-plane based proof system has (maximal) rank n on a particular family of instances, then any cutting-plane proof system in our class has rank Ω(n/ log n) for this family. This shows that the rank complexity of worst-case instances is intrinsic to the problem; it does not depend on specific cutting-plane proof systems, except for log factors. We also construct a new cutting-plane proof system that hasworst-case rankO(n/ log n) for any polytopewithout integral points, implying that our universal lower bound is essentially tight.