Gomory-Chvátal cutting planes and the elementary closure of Polyhedra

Abstract

The elementary closure P'; of a polyhedrom P is the intersection of P with all its Gomory-Chvátal cutting planes. P'; is a rational polyhedron provided that P is rational. The Chvátal-Gomory procedure is the iterative application of the elementary closure operation to P. The Chvátal rank is the minimal number of iterations needed to obtain P_I. It is always finite, but already in |R² one can construct polytopes of arbitrary large Chvátal rank. We show that the Chvátal rank of polytopes contained in the n-dimensional 0/1 cube is O(n² log n) and prove the lower bound (1+E) n, for some E> 0. We show that the separation problem for the elementary closure of a rational polyhedron is NP-hard. This solves a problem posed by Schrijver. Last we consider the elementary closure in fixed dimension. the known bounds for the number of inequalities defining P'; are exponential, even fixed dimension. We show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension. Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone from this polynomial description in fixed dimension with a maximal degree of violation in a natural sense.