Document Open Access Logo

On Connections Between k-Coloring and Euclidean k-Means

Authors Enver Aman, Karthik C. S. , Sharath Punna

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2024.9.pdf
  • Filesize: 0.82 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity classes
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • k-means
  • k-minsum
  • Euclidean space
  • fine-grained complexity

Metrics

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

Abstract

In the Euclidean k-means problems we are given as input a set of n points in ℝ^d and the goal is to find a set of k points C ⊆ ℝ^d, so as to minimize the sum of the squared Euclidean distances from each point in P to its closest center in C. In this paper, we formally explore connections between the k-coloring problem on graphs and the Euclidean k-means problem. Our results are as follows:  
- For all k ≥ 3, we provide a simple reduction from the k-coloring problem on regular graphs to the Euclidean k-means problem. Moreover, our technique extends to enable a reduction from a structured max-cut problem (which may be considered as a partial 2-coloring problem) to the Euclidean 2-means problem. Thus, we have a simple and alternate proof of the NP-hardness of Euclidean 2-means problem. 
- In the other direction, we mimic the O(1.7297ⁿ) time algorithm of Williams [TCS'05] for the max-cut of problem on n vertices to obtain an algorithm for the Euclidean 2-means problem with the same runtime, improving on the naive exhaustive search running in 2ⁿ⋅ poly(n,d) time. 
- We prove similar results and connections as above for the Euclidean k-min-sum problem.

Cite As Get BibTex

Enver Aman, Karthik C. S., and Sharath Punna. On Connections Between k-Coloring and Euclidean k-Means. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.9

Author Details

Enver Aman
  • Rutgers University, Piscataway, NJ, USA
Karthik C. S.
  • Rutgers University, Piscataway, NJ, USA
Sharath Punna
  • Rutgers University, Piscataway, NJ, USA

Funding

This work was supported by a grant from the Simons Foundation, Grant Number 825876, Awardee Thu D. Nguyen and by the National Science Foundation under Grant CCF-2313372.

Acknowledgements

We would like to thank Pasin Manurangsi for pointing us to [Björklund et al., 2007] and informing us that the fast max-sum convolution result of that paper can be used to obtain a 2ⁿ⋅ poly(n,d) runtime algorithm for the k-means problem. Also, we would like to thank Vincent Cohen-Addad for suggesting to us the that 2-min-sum problem might be more naturally connected to the Max-Cut problem. Finally, we would like to thank the anonymous reviewers for helping us improve the presentation of the paper.

References

  1. Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. Seth-based lower bounds for subset sum and bicriteria path. ACM Transactions on Algorithms (TALG), 18(1):1-22, 2022. Google Scholar
  2. Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The hardness of approximation of euclidean k-means. In Lars Arge and János Pach, editors, 31st International Symposium on Computational Geometry, SoCG 2015, June 22-25, 2015, Eindhoven, The Netherlands, volume 34 of LIPIcs, pages 754-767. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.SOCG.2015.754.
  3. Richard Beigel and David Eppstein. 3-coloring in time O(1.3289ⁿ). Journal of Algorithms, 54(2):168-204, 2005. Google Scholar
  4. P. Berkhin. A Survey of Clustering Data Mining Techniques, pages 25-71. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006. URL: https://doi.org/10.1007/3-540-28349-8_2.
  5. Anup Bhattacharya, Ragesh Jaiswal, and Amit Kumar. Faster algorithms for the constrained k-means problem. Theory Comput. Syst., 62(1):93-115, 2018. URL: https://doi.org/10.1007/s00224-017-9820-7.
  6. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets möbius: fast subset convolution. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 67-74, 2007. Google Scholar
  7. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM Journal on Computing, 39(2):546-563, 2009. Google Scholar
  8. Vincent Cohen-Addad, Andreas Emil Feldmann, and David Saulpic. Near-linear time approximation schemes for clustering in doubling metrics. In J. ACM, volume 68, 2021. Google Scholar
  9. Vincent Cohen-Addad and Karthik C. S. Inapproximability of clustering in lp metrics. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 519-539. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00040.
  10. Vincent Cohen-Addad, Karthik C. S., and Euiwoong Lee. On approximability of clustering problems without candidate centers. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2635-2648. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.156.
  11. Vincent Cohen-Addad, Karthik C. S., and Euiwoong Lee. Johnson coverage hypothesis: Inapproximability of k-means and k-median in 𝓁_p-metrics. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 1493-1530. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.63.
  12. Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On problems as hard as CNF-SAT. ACM Trans. Algorithms, 12(3):41:1-41:24, 2016. URL: https://doi.org/10.1145/2925416.
  13. David P Dailey. Uniqueness of colorability and colorability of planar 4-regular graphs are np-complete. Discrete Mathematics, 30(3):289-293, 1980. Google Scholar
  14. Andreas Darmann and Janosch Döcker. On a simple hard variant of not-all-equal 3-sat. Theoretical Computer Science, 815:147-152, 2020. URL: https://doi.org/10.1016/j.tcs.2020.02.010.
  15. Sanjoy Dasgupta and Yoav Freund. Random projection trees for vector quantization. IEEE Transactions on Information Theory, 55(7):3229-3242, 2009. Google Scholar
  16. W Fernandez De La Vega and Claire Kenyon. A randomized approximation scheme for metric max-cut. Journal of computer and system sciences, 63(4):531-541, 2001. Google Scholar
  17. Uriel Feige. Np-hardness of hypercube 2-segmentation. arXiv preprint arXiv:1411.0821, 2014. Google Scholar
  18. Henry Fleischmann, Kyrylo Karlov, Karthik C. S., Ashwin Padaki, and Stepan Zharkov. Inapproximability of maximum diameter clustering for few clusters. CoRR, abs/2312.02097, 2023. URL: https://doi.org/10.48550/arXiv.2312.02097.
  19. Fedor V Fomin, Serge Gaspers, and Saket Saurabh. Improved exact algorithms for counting 3-and 4-colorings. In Computing and Combinatorics: 13th Annual International Conference, COCOON 2007, Banff, Canada, July 16-19, 2007. Proceedings 13, pages 65-74. Springer, 2007. Google Scholar
  20. Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Nidhi Purohit, and Saket Saurabh. Exact exponential algorithms for clustering problems. In Holger Dell and Jesper Nederlof, editors, 17th International Symposium on Parameterized and Exact Computation, IPEC 2022, September 7-9, 2022, Potsdam, Germany, volume 249 of LIPIcs, pages 13:1-13:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.IPEC.2022.13.
  21. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001. Preliminary version in CCC'99. URL: https://doi.org/10.1006/jcss.2000.1727.
  22. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. Preliminary version in FOCS'98. URL: https://doi.org/10.1006/jcss.2001.1774.
  23. Mary Inaba, Naoki Katoh, and Hiroshi Imai. Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering (extended abstract). In Proceedings of the Tenth Annual Symposium on Computational Geometry, Stony Brook, New York, USA, June 6-8, 1994, pages 332-339, 1994. URL: https://doi.org/10.1145/177424.178042.
  24. Robert E Jensen. A dynamic programming algorithm for cluster analysis. Operations research, 17(6):1034-1057, 1969. Google Scholar
  25. Robert Krauthgamer and Ohad Trabelsi. The set cover conjecture and subgraph isomorphism with a tree pattern. In Rolf Niedermeier and Christophe Paul, editors, 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, March 13-16, 2019, Berlin, Germany, volume 126 of LIPIcs, pages 45:1-45:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.STACS.2019.45.
  26. Amit Kumar, Yogish Sabharwal, and Sandeep Sen. Linear-time approximation schemes for clustering problems in any dimensions. J. ACM, 57(2), 2010. Google Scholar
  27. Michael Lampis. Finer tight bounds for coloring on clique-width. SIAM Journal on Discrete Mathematics, 34(3):1538-1558, 2020. URL: https://doi.org/10.1137/19M1280326.
  28. Euiwoong Lee, Melanie Schmidt, and John Wright. Improved and simplified inapproximability for k-means. Inf. Process. Lett., 120:40-43, 2017. URL: https://doi.org/10.1016/j.ipl.2016.11.009.
  29. Stuart Lloyd. Least squares quantization in pcm. IEEE transactions on information theory, 28(2):129-137, 1982. Google Scholar
  30. Meena Mahajan, Prajakta Nimbhorkar, and Kasturi R. Varadarajan. The planar k-means problem is NP-hard. Theor. Comput. Sci., 442:13-21, 2012. Google Scholar
  31. Jaroslav Nešetřil and Svatopluk Poljak. On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae, 26(2):415-419, 1985. Google Scholar
  32. Noah Stephens-Davidowitz and Vinod Vaikuntanathan. Seth-hardness of coding problems. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 287-301. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00027.
  33. Ulrike Von Luxburg. A tutorial on spectral clustering. Statistics and computing, 17:395-416, 2007. Google Scholar
  34. R Ryan Williams. Algorithms and resource requirements for fundamental problems. PhD thesis, Carnegie Mellon University, 2007. Google Scholar
  35. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2-3):357-365, 2005. Google Scholar
  36. Virginia Vassilevska Williams. Hardness of easy problems: Basing hardness on popular conjectures such as the strong exponential time hypothesis (invited talk). In 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, September 16-18, 2015, Patras, Greece, pages 17-29, 2015. URL: https://doi.org/10.4230/LIPIcs.IPEC.2015.17.
  37. Virginia Vassilevska Williams. Fine-grained algorithms and complexity (invited talk). In 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, pages 3:1-3:1, 2016. URL: https://doi.org/10.4230/LIPIcs.STACS.2016.3.
  38. Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In Proc. Int. Cong. of Math., volume 3, pages 3431-3472, 2018. Google Scholar
  39. Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. New bounds for matrix multiplication: from alpha to omega. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3792-3835. SIAM, 2024. Google Scholar
  40. Xindong Wu, Vipin Kumar, J. Ross Quinlan, Joydeep Ghosh, Qiang Yang, Hiroshi Motoda, Geoffrey J. McLachlan, Angus F. M. Ng, Bing Liu, Philip S. Yu, Zhi-Hua Zhou, Michael S. Steinbach, David J. Hand, and Dan Steinberg. Top 10 algorithms in data mining. Knowl. Inf. Syst., 14(1):1-37, 2008. URL: https://doi.org/10.1007/s10115-007-0114-2.
  41. Or Zamir. Breaking the 2ⁿ barrier for 5-coloring and 6-coloring. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. 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