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

We study the complexity of computing the mixed-integer hull conv(P ∩ Z ×R) 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 P = conv(V ) and n fixed, we compute a vertex description of the mixed-integer hull in polynomial time and give bounds on the number of vertices of the mixed integer hull. Keywords— Mixed-integer hull, polyhedron, mixed-integer concave minimization