On the hardness of computing intersection, union and Minkowski sum of polytopes

For polytopes P(1),P(2) subset of R(d) , we consider the intersection P(1) boolean AND P(2), the convex hull of the union CH(P(1) boolean OR P(2)), and the Minkowski sum P(1)+ P(2). For the Minkowski sum, we prove that enumerating the facets of P(1) + P(2) is NP-hard if P(1) and P(2) are specified by facets, or if P(1) is specified by vertices and P(2) is a polyhedral cone specified by facets. For the 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.