A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
Minkowski addition of polytopes: computational complexity and applications to Gro¨bner bases
SIAM Journal on Discrete Mathematics
A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
Efficient enumeration of the vertices of polyhedra associated with network LP's
Mathematical Programming: Series A and B
The vertex set of a 0/1-polytope is strongly P-enumerable
Computational Geometry: Theory and Applications
Computational Geometry: Theory and Applications
Generating all vertices of a polyhedron is hard
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
f-Vectors of Minkowski Additions of Convex Polytopes
Discrete & Computational Geometry
Convexity recognition of the union of polyhedra
Computational Geometry: Theory and Applications
Proceedings of the twenty-fourth annual symposium on Computational geometry
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For polytopes P,Q ⊂ Rd we consider the intersection P ∪ Q; the convex hull of the union CH(P ∪ Q); and the Minkowski sum P+Q. We prove that given rational H-polytopes P1,P2,Q it is impossible to verify in polynomial time whether Q=P1+P2, unless P=NP. In particular, this shows that there is no output sensitive polynomial algorithm to compute the facets of the Minkowski sum of two arbitrary H-polytopes even if we consider only rational polytopes. Since the convex hull of the union and the intersection of two polytopes relate naturally to the Minkowski sum via the Cayley trick and polarity, similar hardness results follow for these operations as well.