Characterizing Polytopes Contained in the $0/1$-Cube with Bounded Chvátal-Gomory Rank

Let S ⊆ {0, 1} and R be any polytope contained in [0, 1] with R ∩ {0, 1} = S. We prove that R has bounded Chvátal-Gomory rank (CG-rank) provided that S has bounded pitch and bounded gap, where the pitch is the minimum integer p such that all p-dimensional faces of the 0/1-cube have a nonempty intersection with S, and the gap is a measure of the size of the facet coefficients of conv(S). Let H [S̄] denote the subgraph of the n-cube induced by the vertices not in S. We prove that if H [S̄] does not contain a subdivision of a large complete graph, then both the pitch and the gap are bounded. By our main result, this implies that the CG-rank of R is bounded as a function of the treewidth of H [S̄]. We also prove that if S has pitch 3, then the CG-rank of R is always bounded. Both results generalize a recent theorem of Cornuéjols and Lee [8], who proved that the CG-rank is always bounded if the treewidth of H [S̄] is at most 2. Finally, we complement these results by proving that 0/1-polytopes P = conv(S) in R admit extended formulations whose size is bounded in terms of the pitch and the depth D of any circuit deciding membership in S. Our bound is polynomial in n whenever the pitch is constant and D is logarithmic in n.