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

We study the complexity of computing the mixed-integer hull conv(P∩(Zn×Rd)) of a polyhedron PP. Given an inequality description, with one integer variable, the mixed-integer hull can have exponentially many vertices and facets in dd. For n,dn,d fixed, we give an algorithm to find the mixed-integer hull in polynomial time. Given a finite set V⊆Qn+dV⊆Qn+d, with nn 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.