Approximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects

  • Karl Bringmann
  • Tobias Friedrich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)


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. 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
  2. 2.
    Bárány, I., Füredi, Z.: Computing the volume is difficult. Discrete & Computational Geometry 2, 319–326 (1986); Announced at STOC 1986MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bringmann, K., Friedrich, T.: Approximating the volume of unions and intersections of high-dimensional geometric objects (2008),
  4. 4.
    Chan, T.M.: Semi-online maintenance of geometric optima and measures. SIAM J. Comput. 32, 700–716 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. 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
  6. 6.
    Dyer, M.E., Frieze, A.M.: On the complexity of computing the volume of a polyhedron. SIAM J. Comput. 17, 967–974 (1988)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dyer, M.E., Frieze, A.M., Kannan, R.: A random polynomial time algorithm for approximating the volume of convex bodies. J. ACM 38, 1–17 (1991); Announced at STOC 1989MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fredman, M.L., Weide, B.W.: On the complexity of computing the measure of ∪ [a i, b i]. Commun. ACM 21, 540–544 (1978)MathSciNetCrossRefMATHGoogle Scholar
  9. 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
  10. 10.
    Kannan, R., Lovász, L., Simonovits, M.: Random walks and an O *(n 5) volume algorithm for convex bodies. Random Struct. Algorithms 11, 1–50 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 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
  12. 12.
    Karp, R.M., Luby, M.: Monte-carlo algorithms for the planar multiterminal network reliability problem. J. Complexity 1, 45–64 (1985)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Karp, R.M., Luby, M., Madras, N.: Monte-carlo approximation algorithms for enumeration problems. J. Algorithms 10, 429–448 (1989)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Khachiyan, L.G.: The problem of calculating the volume of a polyhedron is enumerably hard. Russian Mathematical Surveys 44, 199–200 (1989)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Klee, V.: Can the measure of ∪ [a i, b i] be computed in less than O(n logn) steps? American Mathematical Monthly 84, 284–285 (1977)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lovász, L., Vempala, S.: Simulated annealing in convex bodies and an O *(n 4) volume algorithm. J. Comput. Syst. Sci. 72, 392–417 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Overmars, M.H., Yap, C.-K.: New upper bounds in Klee’s measure problem. SIAM J. Comput. 20, 1034–1045 (1991); Announced at FOCS 1988MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Roth, D.: On the hardness of approximate reasoning. Artif. Intell. 82, 273–302 (1996)MathSciNetCrossRefGoogle Scholar
  19. 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
  20. 20.
    van Leeuwen, J., Wood, D.: The measure problem for rectangular ranges in d-space. J. Algorithms 2, 282–300 (1981)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans. Evolutionary Computation 3, 257–271 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Karl Bringmann
    • 1
  • Tobias Friedrich
    • 2
    • 3
  1. 1.Universität des SaarlandesSaarbrückenGermany
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany
  3. 3.International Computer Science InstituteBerkeleyUSA

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