Note on the complexity of the mixed-integer hull of a polyhedron

Abstract

We study the complexity of computing the mixed-integer hull conv(P (Zn x Rd)) of a polyhedron P. Given an inequality description, with one integer variable, the mixed-integer hull can have exponentially many vertices and facets in d. For n; d fixed, we give an algorithm to find the mixed-integer hull in polynomial time. Given a finite set V Qn+d, with n fixed, we compute a vertex description of the mixed-integer hull of conv(V ) in polynomial time and give bounds on the number of vertices of the mixed-integer hull.