On the Hardness of Computing Intersection, Union and Minkowski Sum of Polytopes

For polytopes P1, P2 ⊂ R we consider the intersection P1 ∩ P2, the convex hull of the union CH(P1 ∪ P2), and the Minkowski sum P1 + P2. For Minkowski sum we prove that enumerating the facets of P1+P2 is NPhard if P1 and P2 are specified by facets, or if P1 is specified by vertices and P2 is a polyhedral cone specified by facets. For intersection we prove that computing the facets or the vertices of the intersection of two polytopes is NP-hard if one of them is given by vertices and the other by facets. Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NP-hard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NP-complete .