Hard enumeration problems in geometry and combinatorics
SIAM Journal on Algebraic and Discrete Methods
On the complexity of computing the volume of a polyhedron
SIAM Journal on Computing
Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
How good are convex hull algorithms?
Computational Geometry: Theory and Applications
Approximating the centroid is hard
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Generating All Vertices of a Polyhedron Is Hard
Discrete & Computational Geometry
On the Hardness of Computing Intersection, Union and Minkowski Sum of Polytopes
Discrete & Computational Geometry
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Let P be an H-polytope in R^d with vertex set V. The vertex centroid is defined as the average of the vertices in V. We first prove that computing the vertex centroid of anH-polytope, or even just checking whether it lies in a given halfspace, is #P-hard. We also consider the problem of approximating the vertex centroid by finding a point within an @e distance from it and prove this problem to be #P-easy in the sense that it can be solved efficiently using an oracle for some #P-complete problem. In particular, we show that given an oracle for counting the number of vertices of an H-polytope, one can approximate the vertex centroid in polynomial time. Counting the number of vertices of a polytope defined as the intersection of halfspaces is known to be #P-complete. We also show that any algorithm approximating the vertex centroid to any ''sufficiently'' non-trivial (for example constant) distance, can be used to construct a fully polynomial-time approximation scheme for approximating the centroid and also an output-sensitive polynomial algorithm for the Vertex Enumeration problem. Finally, we show that for unbounded polyhedra the vertex centroid cannot be approximated to a distance of d^1^2^-^@d for any fixed constant @d0 unless P=NP.