Abstract : Given a polytope $P\subseteq\R^n$, the Chvátal-Gomory procedure computes iteratively the integer hull $P_I$ of $P$. The Chvátal rank of $P$ is the minimal number of iterations needed to obtain $P_I$. It is always finite, but already the Chvátal rank of polytopes in $\R^2$ 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 Chvátal rank of any polytope $P\subseteq [0,1]^n$ is $\mbox{O}(n^3 \log n)$ and prove the linear upper and lower bound $n$ for the case $P\cap \Z^n = \emptyset$.
Alexander Bockmayr, Friedrich Eisenbrand, Mark Hartmann, Andreas S. Schulz. On the Chvátal Rank of Polytopes in the 0/1 Cube. Discrete Applied Mathematics, Elsevier, 1999, 98 (1-2), pp.21-27. ⟨inria-00099018⟩