On the Chvátal Rank of Certain Inequalities

The Chvatal rank of an inequality ax ≤ b with integral components and valid for the integral hull of a polyhedron P, is the minimum number of rounds of Gomory-Chvatal cutting planes needed to obtain the given inequality. The Chvatal rank is at most one if b is the integral part of the optimum value z(a) of the linear program max{ax : x ∈ P}. We show that, contrary to what was stated or implied by other authors, the converse to the latter statement, namely, the Chvatal rank is at least two if b is less than the integral part of z(a), is not true in general. We establish simple conditions for which this implication is valid, and apply these conditions to several classes of facet-inducing inequalities for travelling salesman polytopes.

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