Computer Science – Computational Geometry
Scientific paper
2008-09-04
Computational Geometry: Theory and Applications, Vol. 43, No. 6-7, pages 601-610, 2010
Computer Science
Computational Geometry
16 pages, To appear in Computational Geometry - Theory and Applications
Scientific paper
10.1016/j.comgeo.2010.03.004
We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects where one can: (1) test whether a given point lies inside the object, (2) sample a point uniformly, (3) calculate the volume of the object in polynomial time. All three oracles can be weak, that is, just approximate. This implies that Klee's measure problem and the hypervolume indicator can be approximated efficiently even though they are #P-hard and hence cannot be solved exactly in time polynomial in the number of dimensions unless P=NP. Our algorithm also allows to approximate efficiently the volume of the union of convex bodies given by weak membership oracles. For the analogous problem of the intersection of high-dimensional geometric objects we prove #P-hardness for boxes and show that there is no multiplicative polynomial-time $2^{d^{1-\epsilon}}$-approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time $\epsilon$-approximation.
Bringmann Karl
Friedrich Tobias
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