Document Open Access Logo

Facility Location on High-Dimensional Euclidean Spaces

Authors Euiwoong Lee, Kijun Shin

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2025.70.pdf
  • Filesize: 1.62 MB
  • 21 pages

Document Identifiers

Author Details

Euiwoong Lee
  • University of Michigan, Ann Arbor, MI, USA
Kijun Shin
  • Seoul National University, South Korea

Funding

  • Lee, Euiwoong: Supported by organization NSF CCF-2236669 and a gift from Google.

Cite As Get BibTex

Euiwoong Lee and Kijun Shin. Facility Location on High-Dimensional Euclidean Spaces. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 70:1-70:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.70

Abstract

Recent years have seen great progress in the approximability of fundamental clustering and facility location problems on high-dimensional Euclidean spaces, including k-Means and k-Median. While they admit strictly better approximation ratios than their general metric versions, their approximation ratios are still higher than the hardness ratios for general metrics, leaving the possibility that the ultimate optimal approximation ratios will be the same between Euclidean and general metrics. Moreover, such an improved algorithm for Euclidean spaces is not known for Uncapaciated Facility Location (UFL), another fundamental problem in the area.
In this paper, we prove that for any γ ≥ 1.6774 there exists ε > 0 such that Euclidean UFL admits a (γ, 1 + 2e^{-γ} - ε)-bifactor approximation algorithm, improving the result of Byrka and Aardal [Byrka and Aardal, 2010]. Together with the (γ, 1 + 2e^{-γ}) NP-hardness in general metrics, it shows the first separation between general and Euclidean metrics for the aforementioned basic problems. We also present an (α_Li - ε)-(unifactor) approximation algorithm for UFL for some ε > 0 in Euclidean spaces, where α_Li ≈ 1.488 is the best-known approximation ratio for UFL by Li [Li, 2013].

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
Keywords
  • Approximation Algorithms
  • Clustering
  • Facility Location

Metrics

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

Related Versions

References

  1. Sara Ahmadian, Ashkan Norouzi-Fard, Ola Svensson, and Justin Ward. Better guarantees for k-means and euclidean k-median by primal-dual algorithms. SIAM Journal on Computing, 49(4):FOCS17-97, 2019. URL: https://doi.org/10.1137/18M1171321.
  2. Per Austrin, Subhash Khot, and Muli Safra. Inapproximability of vertex cover and independent set in bounded degree graphs. Theory of Computing, 7(1):27-43, 2011. URL: https://doi.org/10.4086/TOC.2011.V007A003.
  3. Jaroslaw Byrka and Karen Aardal. An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. SIAM Journal on Computing, 39(6):2212-2231, 2010. URL: https://doi.org/10.1137/070708901.
  4. Vincent Cohen-Addad, Hossein Esfandiari, Vahab Mirrokni, and Shyam Narayanan. Improved approximations for euclidean k-means and k-median, via nested quasi-independent sets. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1621-1628, 2022. URL: https://doi.org/10.1145/3519935.3520011.
  5. Vincent Cohen-Addad, Andreas Emil Feldmann, and David Saulpic. Near-linear time approximation schemes for clustering in doubling metrics. Journal of the ACM (JACM), 68(6):1-34, 2021. URL: https://doi.org/10.1145/3477541.
  6. Vincent Cohen-Addad and CS Karthik. Inapproximability of clustering in lp metrics. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pages 519-539. IEEE, 2019. Google Scholar
  7. Vincent Cohen-Addad, Karthik C S, and Euiwoong Lee. Johnson coverage hypothesis: Inapproximability of k-means and k-median in 𝓁_p-metrics. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1493-1530. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.63.
  8. Vincent Cohen-Addad Viallat, Fabrizio Grandoni, Euiwoong Lee, and Chris Schwiegelshohn. Breaching the 2 lmp approximation barrier for facility location with applications to k-median. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 940-986. SIAM, 2023. Google Scholar
  9. Kishen N Gowda, Thomas Pensyl, Aravind Srinivasan, and Khoa Trinh. Improved bi-point rounding algorithms and a golden barrier for k-median. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 987-1011. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH38.
  10. Fabrizio Grandoni, Rafail Ostrovsky, Yuval Rabani, Leonard J Schulman, and Rakesh Venkat. A refined approximation for euclidean k-means. Information Processing Letters, 176:106251, 2022. URL: https://doi.org/10.1016/J.IPL.2022.106251.
  11. Sudipto Guha and Samir Khuller. Greedy strikes back: Improved facility location algorithms. Journal of algorithms, 31(1):228-248, 1999. URL: https://doi.org/10.1006/JAGM.1998.0993.
  12. Kamal Jain, Mohammad Mahdian, and Amin Saberi. A new greedy approach for facility location problems. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 731-740, 2002. URL: https://doi.org/10.1145/509907.510012.
  13. Kamal Jain and Vijay V Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. Journal of the ACM (JACM), 48(2):274-296, 2001. URL: https://doi.org/10.1145/375827.375845.
  14. Euiwoong Lee and Kijun Shin. Facility location on high-dimensional euclidean spaces, 2024. URL: https://arxiv.org/abs/2501.18105.
  15. Shi Li. A 1.488 approximation algorithm for the uncapacitated facility location problem. Information and Computation, 222:45-58, 2013. URL: https://doi.org/10.1016/J.IC.2012.01.007.
  16. Mohammad Mahdian, Yinyu Ye, and Jiawei Zhang. A 1.52-approximation algorithm for the uncapacitated facility location problem. In Proc. of APPROX, pages 229-242, 2002. Google Scholar
  17. Robert Alexander Rankin. The closest packing of spherical caps in n dimensions. Glasgow Mathematical Journal, 2(3):139-144, 1955. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact