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Abstract

Given a polytope $P \subseteq \mathbb{R}^n$, the Chv\'atal-Gomory procedure computes iteratively the integer hull $P_I$ of $P$. The Chv\'atal rank of $P$ is the minimal number of iterations needed to obtain $P_I$. It is always finite, but already the Chv\'atal rank of polytopes in $\mathbb{R}^2$ can be arbitrarily large. In this paper, we study polytopes in the 0/1~cube, which are of particular interest in combinatorial optimization. We show that the Chv\'atal rank of a polytope $P \subseteq [0,1]^n $ in the 0/1~cube is at most $6 n^3 \log n$ and prove the linear upper and lower bound $n$ for the case $P\cap \mathbb{Z}^n = \emptyset$.