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
Generating all vertices of a polyhedron is hard
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Approximating the centroid is hard
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
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Let $\mathcal{P}$ be an $\mathcal{H}$-polytope in 驴 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 an $\mathcal{H}$-polytope, or even just checking whether it lies in a given halfspace, are #P-hard. We also consider the problem of approximating the vertex centroid by finding a point within an 驴 distance from it and prove this problem to be #P-easy by showing that given an oracle for counting the number of vertices of an $\mathcal{H}$-polytope, one can approximate the vertex centroid in polynomial time. 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 can not be approximated to a distance of $d^{\frac{1}{2}-\delta}$ for any fixed constant 驴 0.