Characterizing Polytopes Contained in the $0/1$-Cube with Bounded Chv\'atal-Gomory Rank

TL;DR

This paper establishes conditions under which polytopes within the 0/1-cube have bounded Chvátal-Gomory rank, linking geometric properties of the set S and the structure of the induced subgraph to the rank.

Contribution

It introduces bounds on the CG-rank based on notch, gap, and the treewidth of the induced subgraph, generalizing previous results for low treewidth.

Findings

Bounded CG-rank when S has bounded notch and gap.

CG-rank is bounded if the induced subgraph has bounded treewidth.

If S has notch 3, the CG-rank is always bounded.

Abstract

Let $S \subseteq \{0,1\}^n$ and $R$ be any polytope contained in $[0,1]^n$ with $R \cap \{0,1\}^n = S$. We prove that $R$ has bounded Chv\'atal-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 $\mathsf{conv}(S)$. 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 both 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 Cornu\'ejols and Lee, who proved that the CG-rank is bounded by a constant if the treewidth of $H[\bar{S}]$ is at most $2$.