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 Chv´ atal rank of the polyhedron. It is well-known that the Chv´ atal 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 3n2lg n. This improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(n3 lgn). 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 + ε)nfor some ε > 0; the best known lower bound was n.