# Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube

## Abstract

Gomory’s and Chvátal’s cutting-plane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvátal rank of the polyhedron. It is well-known that the Chvátal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its integer hull is a 0/1-polytope. We prove that the Chvátal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, the rank of any polytope contained in the *n*-dimensional 0/1-cube is at most 3*n*^{2} lg *n*. This improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(*n*^{3} lg *n*).

Moreover, we refine this result by showing that the rank of any polytope in the 0/1-cube that is defined by inequalities with small coefficients is O(*n*). The latter observation explains why for most cutting planes derived in polyhedral studies of several popular combinatorial optimization problems only linear growth has been observed (see, e.g., [13]); the coefficients of the corresponding inequalities are usually small. Similar results were only known for monotone polyhedra before.

Finally, we provide a family of polytopes contained in the 0/1-cube the Chvátal rank of which is at least (1+*∈*)*n* for some *∈* > 0; the best known lower bound was *n*.

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