Document Open Access Logo

Finer-Grained Reductions in Fine-Grained Hardness of Approximation

Authors Elie Abboud , Noga Ron-Zewi

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.7.pdf
  • Filesize: 0.78 MB
  • 17 pages

Document Identifiers

Author Details

Elie Abboud
  • Department of Computer Science, University of Haifa, Israel
Noga Ron-Zewi
  • Department of Computer Science, University of Haifa, Israel

Funding

  • Abboud, Elie: Research supported in part by ISF grant 735/20, and by the European Union (ERC, ECCC, 101076663). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
  • Ron-Zewi, Noga: Research supported in part by ISF grant 735/20, and by the European Union (ERC, ECCC, 101076663). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

Cite As Get BibTex

Elie Abboud and Noga Ron-Zewi. Finer-Grained Reductions in Fine-Grained Hardness of Approximation. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.7

Abstract

We investigate the relation between δ and ε required for obtaining a (1+δ)-approximation in time N^{2-ε} for closest pair problems under various distance metrics, and for other related problems in fine-grained complexity. 
Specifically, our main result shows that if it is impossible to (exactly) solve the (bichromatic) inner product (IP) problem for vectors of dimension c log N in time N^{2-ε}, then there is no (1+δ)-approximation algorithm for (bichromatic) Euclidean Closest Pair running in time N^{2-2ε}, where δ ≈ (ε/c)² (where ≈ hides polylog factors). This improves on the prior result due to Chen and Williams (SODA 2019) which gave a smaller polynomial dependence of δ on ε, on the order of δ ≈ (ε/c)⁶. Our result implies in turn that no (1+δ)-approximation algorithm exists for Euclidean closest pair for δ ≈ ε⁴, unless an algorithmic improvement for IP is obtained. This in turn is very close to the approximation guarantee of δ ≈ ε³ for Euclidean closest pair, given by the best known algorithm of Almam, Chan, and Williams (FOCS 2016). By known reductions, a similar result follows for a host of other related problems in fine-grained hardness of approximation. 
Our reduction combines the hardness of approximation framework of Chen and Williams, together with an MA communication protocol for IP over a small alphabet, that is inspired by the MA protocol of Chen (Theory of Computing, 2020).

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Fine-grained complexity
  • conditional lower bound
  • fine-grained reduction
  • Approximation algorithms
  • Analysis of algorithms
  • Computational geometry
  • Computational and structural complexity theory

Metrics

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

Related Versions

References

  1. Scott Aaronson and Avi Wigderson. Algebrization: A new barrier in complexity theory. ACM Trans. Comput. Theory, 1(1):2:1-2:54, 2009. Google Scholar
  2. Amir Abboud, Aviad Rubinstein, and Ryan Williams. Distributed pcp theorems for hardness of approximation in p. In FOCS, pages 25-36. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.12.
  3. Amir Abboud, Richard Ryan Williams, and Huacheng Yu. More applications of the polynomial method to algorithm design. In Piotr Indyk, editor, SODA, pages 218-230. SIAM, 2015. Google Scholar
  4. Josh Alman, Timothy Chan, and Ryan Williams. Polynomial representations of threshold functions and algorithmic applications. In FOCS, pages 467-476. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.57.
  5. Josh Alman and Ryan Williams. Probabilistic polynomials and hamming nearest neighbors. In Venkatesan Guruswami, editor, FOCS, pages 136-150. IEEE Computer Society, 2015. URL: http://www.computer.org/csdl/proceedings/focs/2015/8191/00/index.html.
  6. Ethem Alpaydin. Introduction to Machine Learning. MIT Press, Cambridge, Massachusetts, 2014. Google Scholar
  7. Timothy M. Chan and R. Ryan Williams. Deterministic apsp, orthogonal vectors, and more: Quickly derandomizing razborov-smolensky. ACM Trans. Algorithms, 17(1):2:1-2:14, 2021. Google Scholar
  8. Lijie Chen. On the hardness of approximate and exact (bichromatic) maximum inner product. Theory of Computing, 16:1-50, 2020. Google Scholar
  9. Lijie Chen and Ryan Williams. An equivalence class for orthogonal vectors. In SODA, pages 21-40. SIAM, 2019. Google Scholar
  10. Andreas Emil Feldmann, C. S. Karthik, Euiwoong Lee, and Pasin Manurangsi. A survey on approximation in parameterized complexity: Hardness and algorithms. Algorithms, 13(6):146, 2020. Google Scholar
  11. Jiawei Gao, Russell Impagliazzo, Antonina Kolokolova, and Ryan Williams. Completeness for first-order properties on sparse structures with algorithmic applications. ACM Trans. Algorithms, 15(2):23:1-23:35, 2019. Google Scholar
  12. Tomislav Hengl. Finding the right pixel size. Computers and Geosciences, 32(9):1283-1298, 2006. Google Scholar
  13. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci, 62(2):367-375, 2001. Google Scholar
  14. Samir Khuller and Yossi Matias. A simple randomized sieve algorithm for the closest-pair problem. Inf. Comput, 118(1):34-37, April 1995. Google Scholar
  15. C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. Journal of the ACM, 39(4):859-868, 1992. Google Scholar
  16. M. O. Rabin. Probabilistic Algorithms, pages 21-39. Academic Press, NY, 1976. Google Scholar
  17. Aviad Rubinstein. Hardness of approximate nearest neighbor search. In STOC, pages 1260-1268. ACM, 2018. URL: http://dl.acm.org/citation.cfm?id=3188745.
  18. Aviad Rubinstein and Virginia Vassilevska Williams. Seth vs approximation. SIGACT News, 50(4):57-76, 2019. Google Scholar
  19. Kenneth W. Shum, Ilia Aleshnikov, P. Vijay Kumar, Henning Stichtenoth, and Vinay Deolalikar. A low-complexity algorithm for the construction of algebraic-geometric codes better than the gilbert-varshamov bound. IEEE Transactions on Information Theory, 47(6):2225-2241, 2001. URL: https://doi.org/10.1109/18.945244.
  20. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci, 348(2-3):357-365, 2005. Google Scholar
  21. Virginia Vassilevska Williams. Some open problems in fine-grained complexity. SIGACT News, 49(4):29-35, 2018. Google Scholar
  22. Virginia Vassilevska Williams and R. Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5):27:1-27:38, 2018. Google Scholar
  23. Raymond Chi-Wing Wong, Yufei Tao, Ada Wai-Chee Fu, and Xiaokui Xiao. On efficient spatial matching. In VLDB, pages 579-590. ACM, 2007. URL: http://www.vldb.org/conf/2007/papers/research/p579-wong.pdf.
  24. Charles Zahn. Graph-theoretical methods for detecting and describing gestalt clusters. IEEE Transactions on Computers, 20(1):68-86, January 1971. Google Scholar
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