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Efficient Algorithms and Hardness Results for the Weighted k-Server Problem

Authors Anupam Gupta, Amit Kumar, Debmalya Panigrahi

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Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Online Algorithms
  • Weighted k-server
  • Integrality Gap
  • Hardness

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Abstract

In this paper, we study the weighted k-server problem on the uniform metric in both the offline and online settings. We start with the offline setting. In contrast to the (unweighted) k-server problem which has a polynomial-time solution using min-cost flows, there are strong computational lower bounds for the weighted k-server problem, even on the uniform metric. Specifically, we show that assuming the unique games conjecture, there are no polynomial-time algorithms with a sub-polynomial approximation factor, even if we use c-resource augmentation for c < 2. Furthermore, if we consider the natural LP relaxation of the problem, then obtaining a bounded integrality gap requires us to use at least 𝓁 resource augmentation, where 𝓁 is the number of distinct server weights. We complement these results by obtaining a constant-approximation algorithm via LP rounding, with a resource augmentation of (2+ε)𝓁 for any constant ε > 0.
In the online setting, an exp(k) lower bound is known for the competitive ratio of any randomized algorithm for the weighted k-server problem on the uniform metric. In contrast, we show that 2𝓁-resource augmentation can bring the competitive ratio down by an exponential factor to only O(𝓁² log 𝓁). Our online algorithm uses the two-stage approach of first obtaining a fractional solution using the online primal-dual framework, and then rounding it online.

Cite As Get BibTex

Anupam Gupta, Amit Kumar, and Debmalya Panigrahi. Efficient Algorithms and Hardness Results for the Weighted k-Server Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.12

Author Details

Anupam Gupta
  • Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
Amit Kumar
  • Computer Science and Engineering Department, Indian Institute of Technology, Delhi, India
Debmalya Panigrahi
  • Computer Science, Duke University, Durham, NC, USA

Funding

  • Gupta, Anupam: NSF awards CCF-1955785, CCF-2006953, and CCF-2224718.
  • Panigrahi, Debmalya: NSF awards CCF-1750140 (CAREER) and CCF-1955703.

References

  1. Nikhil Ayyadevara and Ashish Chiplunkar. The randomized competitive ratio of weighted k-server is at least exponential. In 29th Annual European Symposium on Algorithms, volume 204 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 9, 11. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2021. Google Scholar
  2. Nikhil Bansal, Niv Buchbinder, Aleksander Madry, and Joseph Naor. A polylogarithmic-competitive algorithm for the k-server problem. J. ACM, 62(5):40:1-40:49, 2015. URL: https://doi.org/10.1145/2783434.
  3. Nikhil Bansal, Niv Buchbinder, and Joseph Naor. Metrical task systems and the k-server problem on HSTs. In Automata, languages and programming. Part I, volume 6198 of Lecture Notes in Comput. Sci., pages 287-298. Springer, Berlin, 2010. URL: https://doi.org/10.1007/978-3-642-14165-2_25.
  4. Nikhil Bansal, Niv Buchbinder, and Joseph Naor. A primal-dual randomized algorithm for weighted paging. J. ACM, 59(4):19, 2012. Google Scholar
  5. Nikhil Bansal, Marek Eliás, Lukasz Jez, and Grigorios Koumoutsos. The (h, k)-server problem on bounded depth trees. ACM Trans. Algorithms, 15(2):28:1-28:26, 2019. URL: https://doi.org/10.1145/3301314.
  6. Nikhil Bansal, Marek Eliáš, and Grigorios Koumoutsos. Weighted k-server bounds via combinatorial dichotomies. In 58th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2017, pages 493-504. IEEE Computer Soc., Los Alamitos, CA, 2017. URL: https://doi.org/10.1109/FOCS.2017.52.
  7. Yair Bartal, Avrim Blum, Carl Burch, and Andrew Tomkins. A polylog(n)-competitive algorithm for metrical task systems. In STOC '97 (El Paso, TX), pages 711-719. ACM, New York, 1999. Google Scholar
  8. S. Ben-David, A. Borodin, R. Karp, G. Tardos, and A. Wigderson. On the power of randomization in on-line algorithms. Algorithmica, 11(1):2-14, 1994. URL: https://doi.org/10.1007/BF01294260.
  9. Avrim Blum, Carl Burch, and Adam Kalai. Finely-competitive paging. In 40th Annual Symposium on Foundations of Computer Science, FOCS '99, 17-18 October, 1999, New York, NY, USA, pages 450-458. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814617.
  10. Allan Borodin and Ran El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, USA, 1998. Google Scholar
  11. Allan Borodin, Nathan Linial, and Michael E. Saks. An optimal on-line algorithm for metrical task system. J. Assoc. Comput. Mach., 39(4):745-763, 1992. URL: https://doi.org/10.1145/146585.146588.
  12. Sébastien Bubeck, Michael B. Cohen, James R. Lee, and Yin Tat Lee. Metrical task systems on trees via mirror descent and unfair gluing. SIAM J. Comput., 50(3):909-923, 2021. URL: https://doi.org/10.1137/19M1237879.
  13. Sébastien Bubeck, Michael B. Cohen, Yin Tat Lee, James R. Lee, and Aleksander Madry. k-server via multiscale entropic regularization. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 3-16. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188798.
  14. Niv Buchbinder, Anupam Gupta, Marco Molinaro, and Joseph (Seffi) Naor. k-servers with a smile: Online algorithms via projections. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 98-116. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.7.
  15. Ashish Chiplunkar and Sundar Vishwanathan. Randomized memoryless algorithms for the weighted and the generalized k-server problems. ACM Trans. Algorithms, 16(1):Art. 14, 28, 2020. URL: https://doi.org/10.1145/3365002.
  16. Christian Coester and James R. Lee. Pure entropic regularization for metrical task systems. Theory Comput., 18:Paper No. 23, 24, 2022. URL: https://doi.org/10.4086/toc.2022.v018a023.
  17. Amos Fiat and Moty Ricklin. Competitive algorithms for the weighted server problem. Theoret. Comput. Sci., 130(1):85-99, 1994. URL: https://doi.org/10.1016/0304-3975(94)90154-6.
  18. D. R. Fulkerson and O. A. Gross. Incidence matrices and interval graphs. Pacific Journal of Mathematics, 15(3):835-855, 1965. Google Scholar
  19. Bala Kalyanasundaram and Kirk Pruhs. Speed is as powerful as clairvoyance. In 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, USA, 23-25 October 1995, pages 214-221. IEEE Computer Society, 1995. URL: https://doi.org/10.1109/SFCS.1995.492478.
  20. Elias Koutsoupias and Christos H. Papadimitriou. On the k-server conjecture. J. ACM, 42(5):971-983, 1995. URL: https://doi.org/10.1145/210118.210128.
  21. James R. Lee. Fusible hsts and the randomized k-server conjecture. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 438-449. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00049.
  22. Mark S. Manasse, Lyle A. McGeoch, and Daniel Dominic Sleator. Competitive algorithms for server problems. J. Algorithms, 11(2):208-230, 1990. URL: https://doi.org/10.1016/0196-6774(90)90003-W.
  23. Daniel Dominic Sleator and Robert Endre Tarjan. Amortized efficiency of list update and paging rules. Commun. ACM, 28(2):202-208, 1985. URL: https://doi.org/10.1145/2786.2793.
  24. Neal E. Young. On-line caching as cache size varies. In Alok Aggarwal, editor, Proceedings of the Second Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms, 28-30 January 1991, San Francisco, California, USA, pages 241-250. ACM/SIAM, 1991. URL: http://dl.acm.org/citation.cfm?id=127787.127832.
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