On the Rank of Cutting-Plane Proof Systems

Abstract

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 includes many well-known methods, 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 has worst-case rank O(n/ log n) for any polytope without integral points, implying that our universal lower bound is essentially tight.

Design and Verify: A New Scheme for Generating Cutting-Planes

“…Now we verify ∂M M. Let n ∈ N be such that M(A n ) = ∅ and M(A n−1 ) = ∅; such an n exists (due to the coordinate rounding property of M we have that M(A 1 ) = ∅ and since the rank of A n with respect to M is in Ω(n/ log(n)) [19], there exists t ∈ N such that M(A t ) = ∅). We claim that ∂M(A n ) = ∅ which implies that ∂M M follows.…”

“…We note here that almost all known classes of cutting-plane schemes such as GC cuts, lift-and-project cuts, split cuts, and N, N 0 , N + are admissible (cf. [19] for more details). Observe that (1) in Section 1 follows from inclusion preservation.…”

“…Recall that we do not prove short verification (property (7.)) for ∂M which is the basis for the lower bound in [19,Corollary 23] for admissible systems. We will show that the lower bound for ∂M is inherited from the original operator M. Let …”

“…In Section 2, we prove basic properties of the V-closure. In order to present these results, we first describe general classes of reasonable cutting-planes, the so called admissible cutting-plane procedures, a machinery developed in [19]. We prove that ∂M is almost admissible, i.e.…”

Integer-empty polytopes in the 0/1-cube with maximal Gomory–Chvátal rank

“…The improvement from O(n 3 log n) in [1] to O(n 2 log n) in [5] as an upper bound on the rank of polytopes in [0, 1] n is a direct consequence of a better upper bound on the rank of certain polytopes in the 0/1-cube that do not contain integral points. It can actually be shown that lower bounds on the rank of polytopes P ⊆ [0, 1] n with P I = ∅ play a crucial role in understanding the rank of any (well-defined) cutting-plane procedure [10]. Moreover, in many cases, the rank of a face F ⊆ P with F I = ∅ induces a lower bound on the rank of P itself.…”