On the Rank of Mixed 0,1 Polyhedra

Abstract

Eisenbrand and Schulz showed recently (IPCO 99) that the maximum Chvátal rank of a polytope in the [0,1]ⁿ cube is bounded above by O(n ² logn) and bounded below by (1 + ∈)n for some ∈ < 0. It is well known that Chvátal’s cuts are equivalent to Gomory’s fractional cuts, which are themselves dominated by Gomory’s mixed integer cuts. What do these upper and lower bounds become when the rank is defined relative to Gomory’s mixed integer cuts? An upper bound of n follows from existing results in the literature. In this note, we show that the lower bound is also equal to n. We relate this result to bounds on the disjunctive rank and on the Lovász-Schrijver rank of polytopes in the [0,1]ⁿ cube. The result still holds for mixed 0,1 polyhedra with nn binary variables.

Supported by NSF grant DMI-9802773 and ONR grant N00014-97-1-0196.