A Subquadratic Bound for Online Bisection

Authors Marcin Bienkowski , Stefan Schmid

Thumbnail PDF


  • Filesize: 0.81 MB
  • 18 pages

Document Identifiers

Author Details

Marcin Bienkowski
  • University of Wrocław, Poland
Stefan Schmid
  • TU Berlin, Germany
  • Weizenbaum Institute, Berlin, Germany

Cite AsGet BibTex

Marcin Bienkowski and Stefan Schmid. A Subquadratic Bound for Online Bisection. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


The online bisection problem is a natural dynamic variant of the classic optimization problem, where one has to dynamically maintain a partition of n elements into two clusters of cardinality n/2. During runtime, an online algorithm is given a sequence of requests, each being a pair of elements: an inter-cluster request costs one unit while an intra-cluster one is free. The algorithm may change the partition, paying a unit cost for each element that changes its cluster. This natural problem admits a simple deterministic O(n²)-competitive algorithm [Avin et al., DISC 2016]. While several significant improvements over this result have been obtained since the original work, all of them either limit the generality of the input or assume some form of resource augmentation (e.g., larger clusters). Moreover, the algorithm of Avin et al. achieves the best known competitive ratio even if randomization is allowed. In this paper, we present the first randomized online algorithm that breaks this natural quadratic barrier and achieves a competitive ratio of Õ(n^{23/12}) without resource augmentation and for an arbitrary sequence of requests.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Bisection
  • Graph Partitioning
  • online balanced Repartitioning
  • online Algorithms
  • competitive Analysis


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Sanjeev Arora, David R. Karger, and Marek Karpinski. Polynomial time approximation schemes for dense instances of NP-hard problems. Journal of Computer and System Sciences, 58(1):193-210, 1999. URL: https://doi.org/10.1006/jcss.1998.1605.
  2. Chen Avin, Marcin Bienkowski, Andreas Loukas, Maciej Pacut, and Stefan Schmid. Dynamic balanced graph partitioning. SIAM Journal on Discrete Mathematics, 34(3):1791-1812, 2020. URL: https://doi.org/10.1137/17M1158513.
  3. Chen Avin, Andreas Loukas, Maciej Pacut, and Stefan Schmid. Online balanced repartitioning. In Proc. 30th Int. Symp. on Distributed Computing (DISC), pages 243-256, 2016. URL: https://doi.org/10.1007/978-3-662-53426-7_18.
  4. Marcin Bienkowski, Martin Böhm, Martin Koutecký, Thomas Rothvoß, Jirí Sgall, and Pavel Veselý. Improved analysis of online balanced clustering. In Proc. 19th Workshop on Approximation and Online Algorithms (WAOA), pages 224-233, 2021. URL: https://doi.org/10.1007/978-3-030-92702-8_14.
  5. Alan Borodin, Nati Linial, and Michael E. Saks. An optimal on-line algorithm for metrical task system. Journal of the ACM, 39(4):745-763, 1992. URL: https://doi.org/10.1145/146585.146588.
  6. Allan Borodin and Ran El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, 1998. Google Scholar
  7. Uriel Feige and Robert Krauthgamer. A polylogarithmic approximation of the minimum bisection. SIAM Journal on Computing, 31(4):1090-1118, 2002. URL: https://doi.org/10.1137/S0097539701387660.
  8. Uriel Feige, Robert Krauthgamer, and Kobbi Nissim. Approximating the minimum bisection size. In Proc. 32nd ACM Symp. on Theory of Computing (STOC), pages 530-536, 2000. URL: https://doi.org/10.1145/335305.335370.
  9. Tobias Forner, Harald Räcke, and Stefan Schmid. Online balanced repartitioning of dynamic communication patterns in polynomial time. In 2nd Symposium on Algorithmic Principles of Computer Systems (APOCS), pages 40-54, 2021. URL: https://doi.org/10.1137/1.9781611976489.4.
  10. Monika Henzinger, Stefan Neumann, Harald Räcke, and Stefan Schmid. Tight bounds for online graph partitioning. In Proc. 32nd ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 2799-2818, 2021. URL: https://doi.org/10.1137/1.9781611976465.166.
  11. Monika Henzinger, Stefan Neumann, and Stefan Schmid. Efficient distributed workload (re-)embedding. In Proc. SIGMETRICS/Performance Joint Int. Conf. on Measurement and Modeling of Computer Systems, pages 43-44, 2019. URL: https://doi.org/10.1145/3309697.3331503.
  12. Robert Krauthgamer. Minimum bisection. In Encyclopedia of Algorithms, pages 1294-1297. Springer, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_231.
  13. Robert Krauthgamer and Uriel Feige. A polylogarithmic approximation of the minimum bisection. SIAM Review, 48(1):99-130, 2006. URL: https://doi.org/10.1137/050640904.
  14. Bohdan S. Majewski and George Havas. The complexity of greatest common divisor computations. In 1st Symposium on Algorithmic Number Theory (ANTS-I), pages 184-193, 1994. URL: https://doi.org/10.1007/3-540-58691-1_56.
  15. Maciej Pacut, Mahmoud Parham, and Stefan Schmid. Brief announcement: Deterministic lower bound for dynamic balanced graph partitioning. In Proc. 39th ACM Symp. on Principles of Distributed Computing (PODC), pages 461-463, 2020. URL: https://doi.org/10.1145/3382734.3405696.
  16. Maciej Pacut, Mahmoud Parham, and Stefan Schmid. Optimal online balanced graph partitioning. In Proc. 40th IEEE Int. Conf. on Computer Communications (INFOCOM), pages 1-9, 2021. URL: https://doi.org/10.1109/INFOCOM42981.2021.9488824.
  17. Harald Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In Proc. 40th ACM Symp. on Theory of Computing (STOC), pages 255-264, 2008. URL: https://doi.org/10.1145/1374376.1374415.
  18. Harald Räcke, Stefan Schmid, and Ruslan Zabrodin. Approximate dynamic balanced graph partitioning. In Proc. 34th ACM Symp. on Parallelism in Algorithms and Architectures (SPAA), pages 401-409, 2022. URL: https://doi.org/10.1145/3490148.3538563.
  19. Rajmohan Rajaraman and Omer Wasim. Improved bounds for online balanced graph re-partitioning. In Proc. 30th European Symp. on Algorithms (ESA), pages 83:1-83:15, 2022. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.83.
  20. Huzur Saran and Vijay V. Vazirani. Finding k cuts within twice the optimal. SIAM Journal on Computing, 24(1):101-108, 1995. URL: https://doi.org/10.1137/S0097539792251730.
  21. Daniel D. Sleator and Robert E. Tarjan. Amortized efficiency of list update and paging rules. Communications of the ACM, 28(2):202-208, 1985. URL: https://doi.org/10.1145/2786.2793.