Document Open Access Logo

Deterministic Maximum Flows in Simple Graphs

Author Tianyi Zhang

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2021.114.pdf
  • Filesize: 0.73 MB
  • 16 pages

Document Identifiers

Author Details

Tianyi Zhang
  • Tsinghua University, Beijing, China

Acknowledgements

I want to thank helpful discussions with my advisor Ran Duan as well as my colleague Shucheng Chi.

Cite AsGet BibTex

Tianyi Zhang. Deterministic Maximum Flows in Simple Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 114:1-114:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.114

Abstract

In this paper we are interested in deterministically computing maximum flows in undirected simple graphs where edges have unit capacities. When the input graph has n vertices and m edges, and the maximum flow is known to be upper bounded by τ as prior knowledge, our algorithm has running time Õ(m + n^{5/3}τ^{1/2}); in the extreme case where τ = Θ(n), our algorithm has running time Õ(n^{2.17}). This always improves upon the previous best deterministic upper bound Õ(n^{9/4}τ^{1/8}) by [Duan, 2013]. Furthermore, when τ ≥ n^{0.67} our algorithm is faster than a classical upper bound of O(m + nτ^{3/2}) by [Karger and Levin, 1998].

Subject Classification

ACM Subject Classification
  • Theory of computation → Network flows
Keywords
  • graph algorithms
  • maximum flows
  • dynamic data structures

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Julia Chuzhoy, Yu Gao, Jason Li, Danupon Nanongkai, Richard Peng, and Thatchaphol Saranurak. A deterministic algorithm for balanced cut with applications to dynamic connectivity, flows, and beyond. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 1158-1167. IEEE, 2020. Google Scholar
  2. Julia Chuzhoy and Thatchaphol Saranurak. Deterministic algorithms for decremental shortest paths via layered core decomposition. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2478-2496. SIAM, 2021. Google Scholar
  3. Efim A Dinic. Algorithm for solution of a problem of maximum flow in networks with power estimation. In Soviet Math. Doklady, volume 11, pages 1277-1280, 1970. Google Scholar
  4. Ran Duan. Breaking the 𝒪(n^2.5) deterministic time barrier for undirected unit-capacity maximum flow. In Proceedings of the twenty-fourth annual ACM-SIAM Symposium on Discrete Algorithms, pages 1171-1179. SIAM, 2013. Google Scholar
  5. Lester Randolph Ford Jr and Delbert Ray Fulkerson. Flows in networks. Princeton university press, 2015. Google Scholar
  6. Andrew V Goldberg and Satish Rao. Beyond the flow decomposition barrier. Journal of the ACM (JACM), 45(5):783-797, 1998. Google Scholar
  7. Andrew V Goldberg and Robert E Tarjan. A new approach to the maximum-flow problem. Journal of the ACM (JACM), 35(4):921-940, 1988. Google Scholar
  8. Jacob Holm, Kristian De Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the ACM (JACM), 48(4):723-760, 2001. Google Scholar
  9. Jacob Holm, Eva Rotenberg, and Christian Wulff-Nilsen. Faster fully-dynamic minimum spanning forest. In Algorithms-ESA 2015, pages 742-753. Springer, 2015. Google Scholar
  10. David R Karger. Using random sampling to find maximum flows in uncapacitated undirected graphs. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 240-249, 1997. Google Scholar
  11. David R Karger and Matthew S Levine. Finding maximum flows in undirected graphs seems easier than bipartite matching. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 69-78, 1998. Google Scholar
  12. David R Karger and Matthew S Levine. Random sampling in residual graphs. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 63-66, 2002. Google Scholar
  13. Tarun Kathuria. A potential reduction inspired algorithm for exact max flow in almost Õ(m^4/3) time. arXiv preprint, 2020. URL: http://arxiv.org/abs/2009.03260.
  14. Yin Tat Lee and Aaron Sidford. Path finding methods for linear programming: Solving linear programs in Õ(√rank) iterations and faster algorithms for maximum flow. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 424-433. IEEE, 2014. Google Scholar
  15. Yang P Liu and Aaron Sidford. Faster divergence maximization for faster maximum flow. arXiv preprint, 2020. URL: http://arxiv.org/abs/2003.08929.
  16. Yang P Liu and Aaron Sidford. Faster energy maximization for faster maximum flow. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 803-814, 2020. Google Scholar
  17. Aleksander Madry. Navigating central path with electrical flows: From flows to matchings, and back. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 253-262. IEEE, 2013. Google Scholar
  18. Aleksander Madry. Computing maximum flow with augmenting electrical flows. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 593-602. IEEE, 2016. Google Scholar
  19. Jan van den Brand, Yin Tat Lee, Yang P Liu, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, and Di Wang. Minimum cost flows, mdps, and ell_1-regression in nearly linear time for dense instances. arXiv e-prints, 2021. URL: http://arxiv.org/abs/2101.05719.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact