For polytopes P 1 , P 2 ⊂ R d we consider the intersection P 1 ∩ P 2 , 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 P 1 and P 2 are specified by facets, or if P 1 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 .
Tiwary REF showed that the deciding whether an H-polytope is equivalent to the convex hull or the Minkowski sum of two H-polytopes is co-NP-complete.
On the hardness of computing intersection, union and minkowski sum of polytopes
DISCRETE & COMPUTATIONAL GEOMETRY
Computer Science Mathematics