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Incremental Approximate Maximum Flow via Residual Graph Sparsification

Authors Gramoz Goranci , Monika Henzinger , Harald Räcke , A. R. Sricharan

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  • Theory of computation → Dynamic graph algorithms
Keywords
  • incremental flow
  • sparsification
  • approximate flow

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Abstract

We give an algorithm that, with high probability, maintains a (1-ε)-approximate s-t maximum flow in undirected, uncapacitated n-vertex graphs undergoing m edge insertions in Õ(m+ n F^*/ε) total update time, where F^{*} is the maximum flow on the final graph. This is the first algorithm to achieve polylogarithmic amortized update time for dense graphs (m = Ω(n²)), and more generally, for graphs where F^* = Õ(m/n).
At the heart of our incremental algorithm is the residual graph sparsification technique of Karger and Levine [SICOMP '15], originally designed for computing exact maximum flows in the static setting. Our main contributions are (i) showing how to maintain such sparsifiers for approximate maximum flows in the incremental setting and (ii) generalizing the cut sparsification framework of Fung et al. [SICOMP '19] from undirected graphs to balanced directed graphs.

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Gramoz Goranci, Monika Henzinger, Harald Räcke, and A. R. Sricharan. Incremental Approximate Maximum Flow via Residual Graph Sparsification. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 91:1-91:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.91

Author Details

Gramoz Goranci
  • Faculty of Computer Science, University of Vienna, Austria
Monika Henzinger
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Harald Räcke
  • Technical University Munich, Germany
A. R. Sricharan
  • Faculty of Computer Science, UniVie Doctoral School Computer Science DoCS, University of Vienna, Austria

Funding

Monika Henzinger and A. R. Sricharan: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (MoDynStruct, No. 101019564) and the Austrian Science Fund (FWF) grant https://www.doi.org/10.55776/Z422, grant https://www.doi.org/10.55776/I5982, and grant https://www.doi.org/10.55776/P33775 with additional funding from the netidee SCIENCE Stiftung, 2020–2024.
  • Räcke, Harald: This project has received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 498605858 and 470029389.

References

  1. Ravindra K Ahuja, Thomas L Magnanti, James B Orlin, and MR Reddy. Applications of network optimization. Handbooks in Operations Research and Management Science, 7:1-83, 1995. URL: https://www.sciencedirect.com/science/article/abs/pii/S0927050705801185.
  2. Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory Comput., 8(1):121-164, 2012. URL: https://doi.org/10.4086/TOC.2012.V008A006.
  3. András A. Benczúr and David R. Karger. Randomized approximation schemes for cuts and flows in capacitated graphs. SIAM J. Comput., 44(2):290-319, 2015. URL: https://doi.org/10.1137/070705970.
  4. Ruoxu Cen, Yu Cheng, Debmalya Panigrahi, and Kevin Sun. Sparsification of directed graphs via cut balance. In International Colloqium on Automata, Languages, and Programming (ICALP), pages 45:1-45:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ICALP.2021.45.
  5. Li Chen, Gramoz Goranci, Monika Henzinger, Richard Peng, and Thatchaphol Saranurak. Fast dynamic cuts, distances and effective resistances via vertex sparsifiers. In Foundations of Computer Science (FOCS), pages 1135-1146. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00109.
  6. Li Chen, Rasmus Kyng, Yang P. Liu, Simon Meierhans, and Maximilian Probst Gutenberg. Almost-linear time algorithms for incremental graphs: Cycle detection, sccs, s-t shortest path, and minimum-cost flow. In Symposium on Theory of Computing (STOC), pages 1165-1173. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649745.
  7. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. In Foundations of Computer Science (FOCS), pages 612-623. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00064.
  8. Søren Dahlgaard. On the hardness of partially dynamic graph problems and connections to diameter. In International Colloqium on Automata, Languages, and Programming (ICALP), pages 48:1-48:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.ICALP.2016.48.
  9. David Durfee, Yu Gao, Gramoz Goranci, and Richard Peng. Fully dynamic effective resistances. CoRR, abs/1804.04038, 2018. URL: https://arxiv.org/abs/1804.04038.
  10. David Durfee, Yu Gao, Gramoz Goranci, and Richard Peng. Fully dynamic spectral vertex sparsifiers and applications. In Symposium on Theory of Computing (STOC), pages 914-925, 2019. URL: https://doi.org/10.1145/3313276.3316379.
  11. Alina Ene, Gary L. Miller, Jakub Pachocki, and Aaron Sidford. Routing under balance. In Symposium on Theory of Computing (STOC), pages 598-611, 2016. URL: https://doi.org/10.1145/2897518.2897654.
  12. Wai Shing Fung, Ramesh Hariharan, Nicholas J. A. Harvey, and Debmalya Panigrahi. A general framework for graph sparsification. SIAM J. Comput., 48(4):1196-1223, 2019. URL: https://doi.org/10.1137/16M1091666.
  13. Yu Gao, Yang P. Liu, and Richard Peng. Fully dynamic electrical flows: Sparse maxflow faster than goldberg-rao. In Foundations of Computer Science, (FOCS), pages 516-527, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00058.
  14. Andrew V. Goldberg and Satish Rao. Beyond the flow decomposition barrier. J. ACM, 45(5):783-797, 1998. URL: https://doi.org/10.1145/290179.290181.
  15. Gramoz Goranci and Monika Henzinger. Efficient data structures for incremental exact and approximate maximum flow. In International Colloqium on Automata, Languages, and Programming (ICALP), pages 69:1-69:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.69.
  16. Gramoz Goranci, Harald Räcke, Thatchaphol Saranurak, and Zihan Tan. The expander hierarchy and its applications to dynamic graph algorithms. In Symposium on Discrete Algorithms (SODA), pages 2212-2228. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.132.
  17. Manoj Gupta and Shahbaz Khan. Simple dynamic algorithms for maximal independent set, maximum flow and maximum matching. In Symposium on Simplicity in Algorithms (SOSA), pages 86-91. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.10.
  18. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Symposium on Theory of Computing (STOC), pages 21-30. ACM, 2015. URL: https://doi.org/10.1145/2746539.2746609.
  19. Monika Rauch Henzinger. A static 2-approximation algorithm for vertex connectivity and incremental approximation algorithms for edge and vertex connectivity. J. Algorithms, 24(1):194-220, 1997. URL: https://doi.org/10.1006/JAGM.1997.0855.
  20. Giuseppe F. Italiano. Amortized efficiency of a path retrieval data structure. Theor. Comput. Sci., 48(3):273-281, 1986. URL: https://doi.org/10.1016/0304-3975(86)90098-8.
  21. David R. Karger and Matthew S. Levine. Fast augmenting paths by random sampling from residual graphs. SIAM J. Comput., 44(2):320-339, 2015. URL: https://doi.org/10.1137/070705994.
  22. Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Symposium on Discrete Algorithms (SODA), pages 217-226. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.16.
  23. Hiroshi Nagamochi and Toshihide Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7(5&6):583-596, 1992. URL: https://doi.org/10.1007/BF01758778.
  24. Richard Peng. Approximate undirected maximum flows in O(mpolylog(n)) time. In Symposium on Discrete Algorithms (SODA), pages 1862-1867. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH130.
  25. Jonah Sherman. Breaking the multicommodity flow barrier for o(vlog n)-approximations to sparsest cut. In Foundations of Computer Science (FOCS), pages 363-372. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/FOCS.2009.66.
  26. Jonah Sherman. Nearly maximum flows in nearly linear time. In Foundations of Computer Science (FOCS), pages 263-269. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.36.
  27. Jonah Sherman. Area-convexity, l_∞ regularization, and undirected multicommodity flow. In Symposium on Theory of Computing (STOC), pages 452-460, 2017. URL: https://doi.org/10.1145/3055399.3055501.
  28. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. SIAM J. Comput., 40(6):1913-1926, 2011. URL: https://doi.org/10.1137/080734029.
  29. Robert Endre Tarjan and Jan van Leeuwen. Worst-case analysis of set union algorithms. J. ACM, 31(2):245-281, 1984. URL: https://doi.org/10.1145/62.2160.
  30. Shang-Hua Teng. The laplacian paradigm: Emerging algorithms for massive graphs. In Theory and Applications of Models of Computation (TAMC), pages 2-14, 2010. URL: https://doi.org/10.1007/978-3-642-13562-0_2.
  31. Jan van den Brand, Li Chen, Rasmus Kyng, Yang P. Liu, Simon Meierhans, Maximilian Probst Gutenberg, and Sushant Sachdeva. Almost-linear time algorithms for decremental graphs: Min-cost flow and more via duality. In Foundations of Computer Science (FOCS), pages 2010-2032. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00120.
  32. Jan van den Brand, Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, Sushant Sachdeva, and Aaron Sidford. Incremental approximate maximum flow on undirected graphs in subpolynomial update time. In Symposium on Discrete Algorithms (SODA), pages 2980-2998. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.106.
  33. Jan van den Brand, Yang P. Liu, and Aaron Sidford. Dynamic maxflow via dynamic interior point methods. In Symposium on Theory of Computing (STOC), pages 1215-1228. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585135.
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