Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
133 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Online Disjoint Set Covers: Randomization is not Necessary (2404.15554v3)

Published 23 Apr 2024 in cs.DS

Abstract: In the online disjoint set covers problem, the edges of a hypergraph are revealed online, and the goal is to partition them into a maximum number of disjoint set covers. That is, n nodes of a hypergraph are given at the beginning, and then a sequence of hyperedges (subsets of [n]) is presented to an algorithm. For each hyperedge, an online algorithm must assign a color (an integer). Once an input terminates, the gain of the algorithm is the number of colors that correspond to valid set covers (i.e., the union of hyperedges that have that color contains all n nodes). We present a deterministic online algorithm that is O(log2 n)-competitive, exponentially improving on the previous bound of O(n) and matching the performance of the best randomized algorithm by Emek et al. [ESA 2019]. For color selection, our algorithm uses a novel potential function, which can be seen as an online counterpart of the derandomization method of conditional probabilities and pessimistic estimators. There are only a few cases where derandomization has been successfully used in the field of online algorithms. In contrast to previous approaches, our result extends to the following new challenges: (i) the potential function derandomizes not only the Chernoff bound, but also the coupon collector's problem, (ii) the value of OPT of the maximization problem is not bounded a priori, and (iii) we do not produce a fractional solution first, but work directly on the input.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (24)
  1. An O(log k)-competitive algorithm for generalized caching. ACM Transactions on Algorithms, 15(1):6:1–6:18, 2019. doi:10.1145/3280826.
  2. The online set cover problem. SIAM Journal on Computing, 39(2):361–370, 2009. doi:10.1137/060661946.
  3. The Probabilistic Method, Second Edition. John Wiley, 2000. doi:10.1002/0471722154.
  4. Power efficient monitoring management in sensor networks. In 2004 IEEE Wireless Communications and Networking Conference, pages 2329–2334. IEEE, 2004. doi:10.1109/WCNC.2004.1311452.
  5. A nearly optimal deterministic online algorithm for non-metric facility location. In Proc. 38th Symp. on Theoretical Aspects of Computer Science (STACS), LIPIcs, pages 14:1–14:17, 2021. doi:10.4230/LIPIcs.STACS.2021.14.
  6. Cover-decomposition and polychromatic numbers. SIAM Journal on Discrete Mathematics, 27(1):240–256, 2013. doi:10.1137/110856332.
  7. An optimal on-line algorithm for metrical task system. Journal of the ACM, 39(4):745–763, 1992. doi:10.1145/146585.146588.
  8. Online Computation and Competitive Analysis. Cambridge University Press, 1998.
  9. The randomized k-server conjecture is false! In Proc. 55th ACM Symp. on Theory of Computing (STOC), pages 581–594. ACM, 2023. doi:10.1145/3564246.3585132.
  10. Metrical task systems on trees via mirror descent and unfair gluing. In Proc. 30th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 89–97. SIAM, 2019. doi:10.1137/1.9781611975482.6.
  11. Online primal-dual algorithms for covering and packing. Mathematics of Operations Research, 34(2):270–286, 2009. doi:10.1287/MOOR.1080.0363.
  12. Restricted strip covering and the sensor cover problem. In Proc. 18th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 1056–1063. SIAM, 2007. doi:10.5555/1283383.1283497.
  13. Improving wireless sensor network lifetime through power aware organization. Wireless Networks, 11(3):333–340, 2005. doi:10.1007/S11276-005-6615-6.
  14. Energy-efficient target coverage in wireless sensor networks. In Proc. 2005 IEEE Int. Conf. on Computer Communications (INFOCOM), pages 1976–1984. IEEE, 2005. doi:10.1109/INFCOM.2005.1498475.
  15. Online disjoint set cover without prior knowledge. In Proc. 27th European Symp. on Algorithms (ESA), LIPIcs, pages 44:1–44:16, 2019. doi:10.4230/LIPICS.ESA.2019.44.
  16. Approximating the domatic number. SIAM Journal on Computing, 32(1):172–195, 2002. doi:10.1137/S0097539700380754.
  17. Optimally decomposing coverings with translates of a convex polygon. Discret. Comput. Geom., 46(2):313–333, 2011. doi:10.1007/S00454-011-9353-9.
  18. Ram P. Gupta. On the chromatic index and the cover index of a multigraph. In Theory and Applications of Graphs, pages 204–215. Springer Berlin Heidelberg, 1978.
  19. The power of clairvoyance for multi-level aggregation and set cover with delay. In Proc. 2023 ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 1594–1610. SIAM, 2023. doi:10.1137/1.9781611977554.CH59.
  20. Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis. Cambridge University Press, USA, 2nd edition, 2017.
  21. The online disjoint set cover problem and its applications. In Proc. 2015 IEEE Int. Conf. on Computer Communications (INFOCOM), pages 1221–1229. IEEE, 2015. doi:10.1109/INFOCOM.2015.7218497.
  22. Optimally approximating the coverage lifetime of wireless sensor networks. IEEE/ACM Transactions on Networking, 25(1):98–111, 2017. doi:10.1109/TNET.2016.2574563.
  23. Prabhakar Raghavan. Probabilistic construction of deterministic algorithms: Approximating packing integer programs. Journal of Computer and System Sciences, 37(2):130–143, 1988. doi:10.1016/0022-0000(88)90003-7.
  24. Neal E. Young. On-line file caching. Algorithmica, 33(3):371–383, 2002. doi:10.1007/S00453-001-0124-5.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now