An algorithm for determining the minimal convex subset that contains all the integer points of a convex polyhedral set

Abstract

An algorithm for determining the minimal convex subset that contains all the integer points of a convex polyhedral set is presented. The algorithm is based on the following result and is in two stages, both of which use a version of integer programming where the variables are represented parametrically [1]: Let P be a set of points in R n ; S p , the convex hull of P and H p , a collection of separating hyperplanes for P. If S′ p is the intersection of the half spaces given by H p , and the vertices of S′ p are in P, then S′ p = S p . In Stage 1, the separating hyperplanes for the original polyhedral set are replaced by a collection of separating hyperplanes for the integer points, with each new hyperplane containing at least n of the integer points that are linearly independent. In Stage 2, additional separating hyperplanes for the integer points (again with each containing at least n of the integer points that are linearly independent) are introduced to the collection so that the intersection of the half spaces given by the collection of hyperplanes has vertices belonging to the set of integer points. This is achieved in a finite number of steps. The final collection of hyperplanes defines the convex hull of the integer points of the original set.