Abstract
k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.82, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm by Agarwal and Procopiuc [Algorithmica 2002] yields an \(O(n\log k)+(1/\epsilon )^{O(2^dk^{1-1/d}\log k)}\)-time \((1+\epsilon )\)-approximation for Euclidean k-center, where d is the dimension. We show for a closely related problem, k-supplier, the double-exponential dependence on dimension is unavoidable if one hopes to have a sub-linear dependence on k in the exponent. We also derive similar algorithmic results to the ones by Agarwal and Procopiuc for both k-center and k-supplier. We use a relatively new tool, called Voronoi separator, which makes our algorithms and analyses substantially simpler. Furthermore we consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a \(2^{O(k\log k)}n^2\) time 3-approximation for NUkC in general metrics, and a \(2^{O((k\log k)/\epsilon )}dn\) time \((1+\epsilon )\)-approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.
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Notes
The hardness result in [6] is under Unique Games Conjecture.
In [1], the authors mentioned the running time is \(O(n\log k)+(k/\epsilon )^{O(dk^{1-1/d})}\) but in fact their STRIP ALGORITHM runs in time \(n^{O(dl^{d-1})}\) for strips of width l and if we substitute l with the term \((d-1)^{1/d}k^{1/d}+2\) of Lemma 2.19 one gets the double-exponential dependence on the dimension.
References
Agarwal, P.K., Procopiuc, C.M.: Exact and approximation algorithms for clustering. Algorithmica 33(2), 201–226 (2002)
Awasthi, P., Charikar, M., Krishnaswamy, R., Sinop, A. K. The hardness of approximation of euclidean k-means. arXiv preprint arXiv:1502.03316 (2015)
Badoiu, M., Clarkson, K. L.: Smaller core-sets for balls. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, Maryland. ACM/SIAM, pp. 801–802 (2003)
Badoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proceedings on 34th Annual ACM Symposium on Theory of Computing. Montréal, Québec, Canada, J. H. Reif, Ed., ACM, pp. 250–257 (2002)
Bandyapadhyay, S.: On perturbation resilience of non-uniform k-center. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2020)
Bhattacharya, A., Goyal, D., Jaiswal, R.: Hardness of approximation of euclidean \( k \)-median. arXiv preprint arXiv:2011.04221 (2020)
Bhattiprolu, V. V. S. P., Har-Peled, S.: Separating a voronoi diagram via local search. In: 32nd International Symposium on Computational Geometry (SoCG 2016). Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2016)
Cabello, S., Giannopoulos, P., Knauer, C., Marx, D., Rote, G.: Geometric clustering: fixed-parameter tractability and lower bounds with respect to the dimension. ACM Trans. Algorithms (TALG) 7(4), 1–27 (2011)
Chakrabarty, D., Goyal, P., Krishnaswamy, R.: The non-uniform k-center problem. ACM Trans. Algorithms (TALG) 16(4), 1–19 (2020)
Chen, R.: On Mentzer’s hardness of the k-center problem on the Euclidean plane
Chitnis, R., Saurabh, N.: Tight lower bounds for approximate & exact k-center in \({\mathbb{R}}^d\). In: 38th International Symposium on Computational Geometry (SoCG 2022), Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2022)
Cohen-Addad, V.: A fast approximation scheme for low-dimensional k-means. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, 2018, A. Czumaj, Ed., SIAM, pp. 430–440 (2018)
Cohen-Addad, V., Lee, E.: Johnson coverage hypothesis: Inapproximability of k-means and k-median in lp-metrics. In: Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, pp. 1493–1530 (2022)
Cygan, M., Fomin, F.V., Kowalik, Ł, Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms, vol. 5. Springer, Berlin (2015)
de Berg, M., Bodlaender, H. L., Kisfaludi-Bak, S., Marx, D., Zanden, T. C. V. d.: A framework for eth-tight algorithms and lower bounds in geometric intersection graphs. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pp. 574–586 (2018)
De La Vega, W. F., Karpinski, M., Kenyon, C., Rabani, Y.: Approximation schemes for clustering problems. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 50–58 (2003)
Dell, H., Husfeldt, T., Marx, D., Taslaman, N., Wahlén, M.: Exponential time complexity of the permanent and the tutte polynomial. ACM Trans. Algorithms (TALG) 10(4), 1–32 (2014)
Feder, T., Greene, D.: Optimal algorithms for approximate clustering. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 434–444 (1988)
Goel, A., Indyk, P., Varadarajan, K. R.: Reductions among high dimensional proximity problems. In: Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, 2001, Washington, DC, USA, S. R. Kosaraju, Ed., ACM/SIAM, pp. 769–778 (2001)
Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)
Har-Peled, S., Mazumdar, S.: On coresets for k-means and k-median clustering. In: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing , pp. 291–300 (2004)
Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. J ACM (JACM) 33(3), 533–550 (1986)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity. J. Comput. Syst. Sci. 63(4), 512–530 (2001)
Jansen, B.: Kernelization for maximum leaf spanning tree with positive vertex weights. In: International Conference on Algorithms and Complexity. Springer, pp. 192–203 (2010)
Johnson, W. B., Lindenstrauss, J.: Extensions of lipschitz mappings into a hilbert space 26. In: Contemporary Mathematics 26 (1984)
Kumar, A., Sabharwal, Y., Sen, S.: Linear-time approximation schemes for clustering problems in any dimensions. J. ACM 57(2), 1–32 (2010)
Mahajan, M., Nimbhorkar, P., Varadarajan, K.R.: The planar k-means problem is np-hard. Theor. Comput. Sci. 442, 13–21 (2012)
Megiddo, N., Supowit, K.J.: On the complexity of some common geometric location problems. SIAM J. Comput. 13(1), 182–196 (1984)
Mentzer, S. G.: Approximability of metric clustering problems. Manuscript https://www.academia.edu/23251714 Approximability of Metric Clustering Problems (1988)
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A preliminary version appeared in AAAI 2022.
Sayan Bandyapadhyay: The work was partly done while the author was a researcher at the University of Bergen, Norway. Supported by the European Research Council (ERC) via grant LOPPRE, reference 819416.
Zachary Friggstad: Supported by an NSERC Discovery Grant and NSERC Discovery Accelerator Supplement Award.
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Bandyapadhyay, S., Friggstad, Z. & Mousavi, R. Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and Variants. Algorithmica (2024). https://doi.org/10.1007/s00453-024-01236-1
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DOI: https://doi.org/10.1007/s00453-024-01236-1