Document Open Access Logo

How to Compute the Volume in Low Dimension?

Authors Arjan Cornelissen , Simon Apers , Sander Gribling

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2025.61.pdf
  • Filesize: 0.93 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Theory of computation → Computational geometry
Keywords
  • Query complexity
  • computational geometry
  • quantum computing
  • volume estimation
  • high-precision limit

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the query complexity of volume estimation to a membership oracle has been a longstanding open question. Most of the previous work focuses on the high-dimensional limit. In this work, we tightly characterize the deterministic, randomized and quantum query complexity of this problem in the high-precision limit, i.e., when the dimension is constant.

Cite As Get BibTex

Arjan Cornelissen, Simon Apers, and Sander Gribling. How to Compute the Volume in Low Dimension?. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.61

Author Details

Arjan Cornelissen
  • Simons Institute, UC Berkeley, CA, USA
  • Université Paris Cité, CNRS, IRIF, Paris, France
Simon Apers
  • Université Paris Cité, CNRS, IRIF, Paris, France
Sander Gribling
  • Tilburg University, The Netherlands

Funding

  • Cornelissen, Arjan: Supported by a Simons-CIQC postdoctoral fellowship through NSF QLCI Grant No. 2016245.
  • Apers, Simon: Supported in part by the European QuantERA project QOPT (ERA-NET Cofund 2022-25), the French PEPR integrated projects EPiQ (ANR-22-PETQ-0007) and HQI (ANR-22-PNCQ-0002), and the French ANR project QUOPS (ANR-22-CE47-0003-01).

References

  1. Pankaj K Agarwal and Sariel Har-Peled. Computing instance-optimal kernels in two dimensions. Discrete & Computational Geometry, pages 1-28, 2024. Google Scholar
  2. Pankaj K Agarwal, Sariel Har-Peled, and Kasturi R Varadarajan. Approximating extent measures of points. Journal of the ACM (JACM), 51(4):606-635, 2004. URL: https://doi.org/10.1145/1008731.1008736.
  3. Andris Ambainis. Quantum lower bounds by quantum arguments. In Proceedings of the 32nd annual ACM symposium on Theory of computing, pages 636-643, 2000. URL: https://doi.org/10.1145/335305.335394.
  4. Nikolay Baldin and Markus Reiß. Unbiased estimation of the volume of a convex body. Stochastic Processes and their Applications, 126(12):3716-3732, 2016. Google Scholar
  5. Imre Bárány and Zoltán Füredi. Computing the volume is difficult. Discret. Comput. Geom., 2:319-326, 1987. URL: https://doi.org/10.1007/BF02187886.
  6. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. Journal of the ACM (JACM), 48(4):778-797, 2001. URL: https://doi.org/10.1145/502090.502097.
  7. Shalev Ben-David and Eric Blais. A tight composition theorem for the randomized query complexity of partial functions. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 240-246. IEEE, 2020. Google Scholar
  8. Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. Contemporary Mathematics, 305:53-74, 2002. Google Scholar
  9. E.M. Bronshteyn and L.D. Ivanov. The approximation of convex sets by polyhedra. Siberian Mathematical Journal, 16(5):852-853, 1975. Google Scholar
  10. Victor-Emmanuel Brunel. Adaptive estimation of convex and polytopal density support. Probability Theory and Related Fields, 164(1):1-16, 2016. Google Scholar
  11. Victor-Emmanuel Brunel. Methods for estimation of convex sets. Statistical Science, 33(4):615-632, 2018. Google Scholar
  12. Shouvanik Chakrabarti, Andrew M. Childs, Shih-Han Hung, Tongyang Li, Chunhao Wang, and Xiaodi Wu. Quantum algorithm for estimating volumes of convex bodies. ACM Transactions on Quantum Computing, 4(3):1-60, 2023. Google Scholar
  13. Timothy M Chan. Faster core-set constructions and data-stream algorithms in fixed dimensions. Computational Geometry, 35(1-2):20-35, 2006. URL: https://doi.org/10.1016/J.COMGEO.2005.10.002.
  14. Yuansi Chen. An almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. Geometric and Functional Analysis, 31:34-61, 2021. Google Scholar
  15. Arjan Cornelissen and Yassine Hamoudi. A sublinear-time quantum algorithm for approximating partition functions. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1245-1264. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH46.
  16. Arjan Cornelissen, Yassine Hamoudi, and Sofiene Jerbi. Near-optimal quantum algorithms for multivariate mean estimation. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 33-43, 2022. Google Scholar
  17. Ben Cousins and Santosh Vempala. Gaussian cooling and O^*(n³) algorithms for volume and Gaussian volume. SIAM Journal on Computing, 47(3):1237-1273, 2018. Google Scholar
  18. R.M. Dudley. Metric entropy of some classes of sets with differentiable boundaries. Journal of Approximation Theory, 10(3):227-236, 1974. Google Scholar
  19. Martin Dyer, Alan Frieze, and Ravi Kannan. A random polynomial-time algorithm for approximating the volume of convex bodies. Journal of the ACM (JACM), 38(1):1-17, 1991. URL: https://doi.org/10.1145/102782.102783.
  20. György Elekes. A geometric inequality and the complexity of computing volume. Discrete & Computational Geometry, 1:289-292, 1986. URL: https://doi.org/10.1007/BF02187701.
  21. Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. Bound on the number of functions that can be distinguished with k quantum queries. Physical Review A, 60(6):4331, 1999. Google Scholar
  22. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization. Springer-Verlag Berlin Heidelberg, 1 edition, 1988. Google Scholar
  23. Sariel Har-Peled and Mitchell Jones. Proof of Dudley’s convex approximation. arXiv preprint arXiv:1912.01977, 2019. URL: https://arxiv.org/abs/1912.01977.
  24. Arun Jambulapati, Yin Tat Lee, and Santosh S. Vempala. A slightly improved bound for the KLS constant. arXiv preprint arXiv:2208.11644, 2022. URL: https://doi.org/10.48550/arXiv.2208.11644.
  25. He Jia, Aditi Laddha, Yin Tat Lee, and Santosh Vempala. Reducing isotropy and volume to KLS: an O^*(n³ ψ²) volume algorithm. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 961-974, 2021. Google Scholar
  26. Bo'az Klartag. Logarithmic bounds for isoperimetry and slices of convex sets. arXiv preprint arXiv:2303.14938, 2023. Google Scholar
  27. Bo’az Klartag and Joseph Lehec. Bourgain’s slicing problem and KLS isoperimetry up to polylog. Geometric and Functional Analysis, 32(5):1134-1159, 2022. Google Scholar
  28. László Lovász and Santosh Vempala. Simulated annealing in convex bodies and an O^*(n⁴) volume algorithm. Journal of Computer and System Sciences, 72(2):392-417, 2006. URL: https://doi.org/10.1016/J.JCSS.2005.08.004.
  29. Ashwin Nayak and Felix Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of the thirty-first annual ACM symposium on Theory of computing, pages 384-393, 1999. URL: https://doi.org/10.1145/301250.301349.
  30. Michael A. Nielsen and Isaac L. Chuang. Quantum computation and quantum information. Cambridge University Press, 2010. Google Scholar
  31. Luis Rademacher and Santosh Vempala. Dispersion of mass and the complexity of randomized geometric algorithms. Advances in Mathematics, 219(3):1037-1069, 2008. Google Scholar
  32. Carsten Schütt. Random polytopes and affine surface area. Mathematische Nachrichten, 170(1):227-249, 1994. Google Scholar
  33. Miklós Simonovits. How to compute the volume in high dimension? Mathematical programming, 97:337-374, 2003. URL: https://doi.org/10.1007/S10107-003-0447-X.
  34. Hai Yu, Pankaj K Agarwal, Raghunath Poreddy, and Kasturi R Varadarajan. Practical methods for shape fitting and kinetic data structures using coresets. Algorithmica, 52:378-402, 2008. URL: https://doi.org/10.1007/S00453-007-9067-9.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact