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Constant-Factor FPT Approximation for Capacitated k-Median

Authors Marek Adamczyk, Jarosław Byrka, Jan Marcinkowski , Syed M. Meesum, Michał Włodarczyk

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Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
  • Theory of computation → Fixed parameter tractability
Keywords
  • K-median
  • Clustering
  • Approximation Algorithms
  • Fixed Parameter Tractability

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Abstract

Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon.

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Marek Adamczyk, Jarosław Byrka, Jan Marcinkowski, Syed M. Meesum, and Michał Włodarczyk. Constant-Factor FPT Approximation for Capacitated k-Median. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ESA.2019.1

Author Details

Marek Adamczyk
  • University of Warsaw, Poland
Jarosław Byrka
  • University of Wrocław, Poland
Jan Marcinkowski
  • University of Wrocław, Poland
Syed M. Meesum
  • University of Wrocław, Poland
Michał Włodarczyk
  • University of Warsaw, Poland

Funding

  • Byrka, Jarosław: Supported by the Polish National Science Centre grant 2015/18/E/ST6/00456.
  • Marcinkowski, Jan: Supported by the Polish National Science Centre grant 2015/18/E/ST6/00456.
  • Meesum, Syed M.: Supported by the Polish National Science Centre grant 2015/18/E/ST6/00456.

References

  1. Karen Aardal, Pieter L van den Berg, Dion Gijswijt, and Shanfei Li. Approximation algorithms for hard capacitated k-facility location problems. European Journal of Operational Research, 242(2):358-368, 2015. Google Scholar
  2. Vijay Arya, Naveen Garg, Rohit Khandekar, Adam Meyerson, Kamesh Munagala, and Vinayaka Pandit. Local search heuristics for k-median and facility location problems. SIAM Journal on Computing, 33(3):544-562, 2004. Google Scholar
  3. Jarosław Byrka, Krzysztof Fleszar, Bartosz Rybicki, and Joachim Spoerhase. Bi-factor approximation algorithms for hard capacitated k-median problems. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 722-736. SIAM, 2015. Google Scholar
  4. Jarosław Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, 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, pages 737-756. SIAM, 2015. Google Scholar
  5. Jaroslaw Byrka, Bartosz Rybicki, and Sumedha Uniyal. An Approximation Algorithm for Uniform Capacitated k-Median Problem with 1+ε Capacity Violation. In Integer Programming and Combinatorial Optimization - 18th International Conference, IPCO 2016, Liège, Belgium, June 1-3, 2016, Proceedings, pages 262-274, 2016. URL: https://doi.org/10.1007/978-3-319-33461-5_22.
  6. Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From gap-ETH to FPT-inapproximability: Clique, dominating set, and more. In Foundations of Computer Science (FOCS), 2017 IEEE 58th Annual Symposium on, pages 743-754. IEEE, 2017. Google Scholar
  7. Moses Charikar, Chandra Chekuri, Ashish Goel, and Sudipto Guha. Rounding via Trees: Deterministic Approximation Algorithms for Group Steiner Trees and k-Median. In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 114-123, 1998. URL: https://doi.org/10.1145/276698.276719.
  8. Moses Charikar, Chandra Chekuri, Ashish Goel, Sudipto Guha, and Serge A. Plotkin. Approximating a Finite Metric by a Small Number of Tree Metrics. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA, pages 379-388, 1998. URL: https://doi.org/10.1109/SFCS.1998.743488.
  9. Moses Charikar, Sudipto Guha, Éva Tardos, and David B Shmoys. A constant-factor approximation algorithm for the k-median problem. In Proceedings of the thirty-first annual ACM symposium on Theory of computing, pages 1-10. ACM, 1999. Google Scholar
  10. Julia Chuzhoy and Yuval Rabani. Approximating k-median with non-uniform capacities. In Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pages 952-958. Society for Industrial and Applied Mathematics, 2005. Google Scholar
  11. Vincent Cohen-Addad and Jason Li. On the Fixed-Parameter Tractability of Capacitated Clustering. In to appear in the 46th International Colloquium on Automata, Languages and Programming (ICALP), 2019. Google Scholar
  12. Marek Cygan, MohammadTaghi Hajiaghayi, and Samir Khuller. LP rounding for k-centers with non-uniform hard capacities. In Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on, pages 273-282. IEEE, 2012. Google Scholar
  13. Artur Czumaj and Christian Sohler. Small space representations for metric min-sum k-clustering and their applications. Theory of Computing Systems, 46(3):416-442, 2010. Google Scholar
  14. H. Gökalp Demirci and Shi Li. Constant Approximation for Capacitated k-Median with (1+ε)-Capacity Violation. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 73:1-73:14, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.73.
  15. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pages 448-455, 2003. URL: https://doi.org/10.1145/780542.780608.
  16. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms. In Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, FOCS '12, pages 470-479, Washington, DC, USA, 2012. IEEE Computer Society. Google Scholar
  17. Anupam Gupta, Euiwoong Lee, and Jason Li. An FPT algorithm beating 2-approximation for k-cut. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2821-2837. Society for Industrial and Applied Mathematics, 2018. Google Scholar
  18. Anupam Gupta, Euiwoong Lee, Jason Li, Pasin Manurangsi, and Michal Wlodarczyk. Losing Treewidth by Separating Subsets. CoRR, abs/1804.01366, 2018. URL: http://arxiv.org/abs/1804.01366.
  19. Madhukar R Korupolu, C Greg Plaxton, and Rajmohan Rajaraman. Analysis of a local search heuristic for facility location problems. Journal of algorithms, 37(1):146-188, 2000. Google Scholar
  20. Euiwoong Lee. Partitioning a graph into small pieces with applications to path transversal. Mathematical Programming, 2018. Preliminary version in SODA 2017. Google Scholar
  21. Shi Li. On uniform capacitated k-median beyond the natural LP relaxation. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 696-707. SIAM, 2015. Google Scholar
  22. Shi Li. Approximating capacitated k-median with (1+ε)k open facilities. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), To Appear, 2016. Google Scholar
  23. Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. In proceedings of the forty-fifth annual ACM symposium on theory of computing, pages 901-910. ACM, 2013. Google Scholar
  24. Jyh-Han Lin and Jeffrey Scott Vitter. ε-Approximations with Minimum Packing Constraint Violation (Extended Abstract). In Proceedings of the Twenty-fourth Annual ACM Symposium on Theory of Computing, STOC '92, pages 771-782, New York, NY, USA, 1992. ACM. URL: https://doi.org/10.1145/129712.129787.
  25. Karthik C. S., Bundit Laekhanukit, and Pasin Manurangsi. On the Parameterized Complexity of Approximating Dominating Set. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 1283-1296, New York, NY, USA, 2018. ACM. URL: https://doi.org/10.1145/3188745.3188896.
  26. Alexander Schrijver. Combinatorial Optimization - Polyhedra and Efficiency. Springer, 2003. Google Scholar
  27. Yicheng Xu, Yong Zhang, and Yifei Zou. A constant parameterized approximation for hard-capacitated k-means. CoRR, abs/1901.04628, 2019. URL: http://arxiv.org/abs/1901.04628.
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