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# FPT Approximation for Capacitated Sum of Radii

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## Cite As

Ragesh Jaiswal, Amit Kumar, and Jatin Yadav. FPT Approximation for Capacitated Sum of Radii. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 65:1-65:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.65

## Abstract

We consider the capacitated clustering problem in general metric spaces where the goal is to identify k clusters and minimize the sum of the radii of the clusters (we call this the Capacitated k-sumRadii problem). We are interested in fixed-parameter tractable (FPT) approximation algorithms where the running time is of the form f(k) ⋅ poly(n), where f(k) can be an exponential function of k and n is the number of points in the input. In the uniform capacity case, Bandyapadhyay et al. recently gave a 4-approximation algorithm for this problem. Our first result improves this to an FPT 3-approximation and extends to a constant factor approximation for any L_p norm of the cluster radii. In the general capacities version, Bandyapadhyay et al. gave an FPT 15-approximation algorithm. We extend their framework to give an FPT (4 + √13)-approximation algorithm for this problem. Our framework relies on a novel idea of identifying approximations to optimal clusters by carefully pruning points from an initial candidate set of points. This is in contrast to prior results that rely on guessing suitable points and building balls of appropriate radii around them. On the hardness front, we show that assuming the Exponential Time Hypothesis, there is a constant c > 1 such that any c-approximation algorithm for the non-uniform capacity version of this problem requires running time 2^Ω(k/polylog(k)).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Facility location and clustering
##### Keywords
• Approximation algorithm
• parameterized algorithm
• clustering

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## References

1. Sayan Bandyapadhyay, William Lochet, and Saket Saurabh. FPT constant-approximations for capacitated clustering to minimize the sum of cluster radii. In Erin W. Chambers and Joachim Gudmundsson, editors, 39th International Symposium on Computational Geometry, SoCG 2023, June 12-15, 2023, Dallas, Texas, USA, volume 258 of LIPIcs, pages 12:1-12:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.SoCG.2023.12.
2. Sayan Bandyapadhyay and Kasturi Varadarajan. Approximate Clustering via Metric Partitioning. In Seok-Hee Hong, editor, 27th International Symposium on Algorithms and Computation (ISAAC 2016), volume 64 of Leibniz International Proceedings in Informatics (LIPIcs), pages 15:1-15:13, Dagstuhl, Germany, 2016. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2016.15.
3. B. Behsaz and M.R. Salavatipour. On minimum sum of radii and diameters clustering. Algorithmica, 73:143-165, 2015. URL: https://doi.org/10.1007/s00453-014-9907-3.
4. Mihai Bădoiu, Sariel Har-Peled, and Piotr Indyk. Approximate clustering via core-sets. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC '02, pages 250-257, New York, NY, USA, 2002. Association for Computing Machinery. URL: https://doi.org/10.1145/509907.509947.
5. Moses Charikar and Rina Panigrahy. Clustering to minimize the sum of cluster diameters. J. Comput. Syst. Sci., 68(2):417-441, 2004. URL: https://doi.org/10.1016/j.jcss.2003.07.014.
6. C.L.Monma and S.Suri. Partitioning points and graphs to minimize the maximum or the sum of diameters. Graph Theory, Combinatorics and Applications, pages 880-912, 1991.
7. Irit Dinur. The pcp theorem by gap amplification. J. ACM, 54(3):12-es, June 2007. URL: https://doi.org/10.1145/1236457.1236459.
8. Srinivas Doddi, Madhav V. Marathe, S. S. Ravi, David Scot Taylor, and Peter Widmayer. Approximation algorithms for clustering to minimize the sum of diameters. Nord. J. Comput., 7(3):185-203, 2000.
9. Zachary Friggstad and Mahya Jamshidian. Improved Polynomial-Time Approximations for Clustering with Minimum Sum of Radii or Diameters. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms (ESA 2022), volume 244 of Leibniz International Proceedings in Informatics (LIPIcs), pages 56:1-56:14, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.56.
10. Matt Gibson, Gaurav Kanade, Erik Krohn, Imran A. Pirwani, and Kasturi Varadarajan. On metric clustering to minimize the sum of radii. Algorithmica, 57:484-498, 2010. URL: https://doi.org/10.1007/s00453-009-9282-7.
11. Matt Gibson, Gaurav Kanade, Erik Krohn, Imran A. Pirwani, and Kasturi Varadarajan. On clustering to minimize the sum of radii. SIAM Journal on Computing, 41(1):47-60, 2012. URL: https://doi.org/10.1137/100798144.
12. Pierre Hansen and Brigitte Jaumard. Cluster analysis and mathematical programming. Math. Program., 79:191-215, 1997. URL: https://doi.org/10.1007/BF02614317.
13. M. Henzinger, D. Leniowski, and C. Mathieu. Dynamic clustering to minimize the sum of radii. Algorithmica, 82:3183-3194, 2020. URL: https://doi.org/10.1007/s00453-020-00721-7.
14. Tanmay Inamdar and Kasturi R. Varadarajan. Capacitated sum-of-radii clustering: An FPT approximation. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), volume 173 of LIPIcs, pages 62:1-62:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.62.