Document Open Access Logo

Polynomial Time Approximation Schemes for Clustering in Low Highway Dimension Graphs

Authors Andreas Emil Feldmann , David Saulpic

PDF
Thumbnail PDF

File

LIPIcs.ESA.2020.46.pdf
  • Filesize: 0.6 MB
  • 22 pages

Document Identifiers

Author Details

Andreas Emil Feldmann
  • Charles University, Prague, Czech Republic
David Saulpic
  • LIP6, Sorbonne Université, Paris, France

Funding

  • Feldmann, Andreas Emil: Supported by the Czech Science Foundation GAČR (grant #19-27871X), and by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004).

Acknowledgements

We thank Vincent Cohen-Addad for helpful discussions.

Cite AsGet BibTex

Andreas Emil Feldmann and David Saulpic. Polynomial Time Approximation Schemes for Clustering in Low Highway Dimension Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 46:1-46:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.46

Abstract

We study clustering problems such as k-Median, k-Means, and Facility Location in graphs of low highway dimension, which is a graph parameter modeling transportation networks. It was previously shown that approximation schemes for these problems exist, which either run in quasi-polynomial time (assuming constant highway dimension) [Feldmann et al. SICOMP 2018] or run in FPT time (parameterized by the number of clusters k, the highway dimension, and the approximation factor) [Becker et al. ESA 2018, Braverman et al. 2020]. In this paper we show that a polynomial-time approximation scheme (PTAS) exists (assuming constant highway dimension). We also show that the considered problems are NP-hard on graphs of highway dimension 1.

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
Keywords
  • Approximation Scheme
  • Clustering
  • Highway Dimension

Metrics

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

Related Versions

References

  1. I. Abraham, A. Fiat, A. V. Goldberg, and R. F. Werneck. Highway dimension, shortest paths, and provably efficient algorithms. In SODA, pages 782-793, 2010. Google Scholar
  2. 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, pages FOCS17-97, 2019. Google Scholar
  3. Sanjeev Arora, Prabhakar Raghavan, and Satish Rao. Approximation schemes for euclidean k-medians and related problems. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC '98, pages 106-113, New York, NY, USA, 1998. ACM. URL: https://doi.org/10.1145/276698.276718.
  4. Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The hardness of approximation of euclidean k-means. In 31st International Symposium on Computational Geometry, SoCG 2015, June 22-25, 2015, Eindhoven, The Netherlands, pages 754-767, 2015. Google Scholar
  5. Yair Bartal and Lee-Ad Gottlieb. A linear time approximation scheme for euclidean TSP. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS, 2013. Google Scholar
  6. A. Becker, P. N. Klein, and D. Saulpic. Polynomial-time approximation schemes for k-center and bounded-capacity vehicle routing in metrics with bounded highway dimension. In ESA, pages 8:1-8:15, 2018. Google Scholar
  7. J. Blum. Hierarchy of transportation network parameters and hardness results. In IPEC, pages 4:1-4:15, 2019. Google Scholar
  8. Vladimir Braverman, Shaofeng H-C Jiang, Robert Krauthgamer, and Xuan Wu. Coresets for clustering in excluded-minor graphs and beyond. arXiv preprint, 2020. URL: http://arxiv.org/abs/2004.07718.
  9. Jaroslaw Byrka, Thomas Pensyl, Bartosz Rybicki, Srinivasan Aravind, and Khoa Trinh. An improved approximation for k-median, and positive correlation in budgeted optimization. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 737-756, 2015. Google Scholar
  10. Vincent Cohen-Addad, Andreas Emil Feldmann, and David Saulpic. Near-linear time approximations schemes for clustering in doubling metrics. In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 540-559, 2019. URL: https://doi.org/10.1109/FOCS.2019.00041.
  11. Vincent Cohen-Addad, Philip N Klein, and Claire Mathieu. Local search yields approximation schemes for k-means and k-median in euclidean and minor-free metrics. SIAM Journal on Computing, 48(2):644-667, 2019. Google Scholar
  12. Y. Disser, A. E. Feldmann, M. Klimm, and J. Könemann. Travelling on graphs with small highway dimension. In Graph-Theoretic Concepts in Computer Science - 45th International Workshop, WG, volume 11789, pages 175-189. Springer, 2019. Google Scholar
  13. A. E. Feldmann. Fixed-parameter approximations for k-center problems in low highway dimension graphs. Algorithmica, 81(3):1031-1052, 2019. Google Scholar
  14. A. E. Feldmann, W. S. Fung, J. Könemann, and I. Post. A (1+ε)-embedding of low highway dimension graphs into bounded treewidth graphs. SIAM Journal on Computing, 47(4):1275-1734, 2018. Google Scholar
  15. Zachary Friggstad, Mohsen Rezapour, and Mohammad R. Salavatipour. Local search yields a PTAS for k-means in doubling metrics. SIAM J. Comput., 48(2):452-480, 2019. URL: https://doi.org/10.1137/17M1127181.
  16. Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS '03, 2003. Google Scholar
  17. Anupam Gupta and Kanat Tangwongsan. Simpler analyses of local search algorithms for facility location. CoRR, abs/0809.2554, 2008. URL: http://arxiv.org/abs/0809.2554.
  18. Stavros G Kolliopoulos and Satish Rao. A nearly linear-time approximation scheme for the euclidean k-median problem. SIAM Journal on Computing, 37(3):757-782, 2007. Google Scholar
  19. Shi Li. A 1.488 approximation algorithm for the uncapacitated facility location problem. Inf. Comput., 222:45-58, 2013. Google Scholar
  20. K. Talwar. Bypassing the embedding: algorithms for low dimensional metrics. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, pages 281-290. ACM, 2004. URL: https://doi.org/10.1145/1007352.1007399.
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 Imprint Privacy Contact