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

Let S ⊆ {0, 1}ⁿ and R be any polytope contained in [0, 1]ⁿ with R ∩ {0, 1}ⁿ = S. We prove that R has bounded Chvátal-Gomory rank (CG-rank) provided that S has bounded notch and bounded gap, where the notch 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 $H[\bar{S}]$ denote the subgraph of the n-cube induced by the vertices not in S. We prove that if $H[\bar{S}]$ does not contain a subdivision of a large complete graph, then the notch 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 $H[\bar{S}]$. We also prove that if S has notch 3, then the CG-rank of R is always bounded. Both results generalize a recent theorem of Cornuéjols and Lee [Cornuéjols G, Lee D (2016) On some polytopes contained in the 0,1 hypercube that have a small Chvátal rank. Louveaux Q, Skutella M, eds. Proc. 18th Internat. Conf. Integer Programming Combinatorial Optim., IPCO ’16 (Springer International, Cham, Switzerland), 300–311], who proved that the CG-rank is bounded by a constant if the treewidth of $H[\bar{S}]$ is at most 2.