The Integer Hull of a Convex Rational Polytope

Abstract

Given A ∈ Z ^m×n and b ∈ Z ^m, we consider the integer program max {c′x′Ax = b; x ∈ N ⁿ} and provide an equivalent and explicit linear program max{ĉ′ q′Mq = r; q ≥ 0}, where M, r, ĉ are easily obtained from A, b, c with no calculation. We also provide an explicit algebraic characterization of the integer hull of the convex polytope P = {x ∈ R ⁿ′Ax = b; x ≥ 0}. All strong valid inequalities can be obtained from the generators of a convex cone whose definition is explicit in terms of M.