Approximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects
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 Open image in new window -approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time ε-approximation.
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- 1.Agarwal, P.K., Kaplan, H., Sharir, M.: Computing the volume of the union of cubes. In: Proc. 23rd annual Symposium on Computational Geometry (SoCG 2007), pp. 294–301 (2007)Google Scholar
- 3.Bringmann, K., Friedrich, T.: Approximating the volume of unions and intersections of high-dimensional geometric objects (2008), http://arxiv.org/abs/0809.0835
- 5.Chan, T.M.: A (slightly) faster algorithm for Klee’s measure problem. In: Proc. 24th ACM Symposium on Computational Geometry (SoCG 2008), pp. 94–100 (2008)Google Scholar
- 9.Halman, N., Klabjan, D., Li, C.-L., Orlin, J.B., Simchi-Levi, D.: Fully polynomial time approximation schemes for stochastic dynamic programs. In: Proc. 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 700–709 (2008)Google Scholar
- 11.Kaplan, H., Rubin, N., Sharir, M., Verbin, E.: Counting colors in boxes. In: Proc. 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 785–794 (2007)Google Scholar
- 19.Suzuki, S., Ibaraki, T.: An average running time analysis of a backtracking algorithm to calculate the measure of the union of hyperrectangles in d dimensions. In: Proc. 16th Canadian Conference on Computational Geometry (CCCG 2004), pp. 196–199 (2004)Google Scholar