Cutting Planes and the Elementary Closure in Fixed Dimension

Abstract

The elementary closure P′ of a polyhedron P is the intersection of P with all its Gomory-Chvátal cutting planes. P′ is a rational polyhedron provided that P is rational. The known bounds for the number of inequalities defining P′ are exponential, even in 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. If P is a simplicial cone, we construct a polytope Q, whose integral elements correspond to cutting planes of P. The vertices of the integer hull QI include the facets of P′. A polynomial upper bound on their number can be obtained by applying a result of Cook et al. (1992). Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone that correspond to vertices of QI.