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Fully Dynamic Spectral Sparsification for Directed Hypergraphs

Authors Sebastian Forster , Gramoz Goranci , Ali Momeni

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LIPIcs.STACS.2026.38.pdf
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ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • Spectral sparsification
  • Dynamic algorithms
  • (Directed) hypergraphs
  • Data structures

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Abstract

There has been a surge of interest in spectral hypergraph sparsification, a natural generalization of spectral sparsification for graphs. In this paper, we present a simple fully dynamic algorithm for maintaining spectral hypergraph sparsifiers of directed hypergraphs. Our algorithm achieves a near-optimal size of O(n² / ε ² log ⁷ m) and amortized update time of O(r² log ³ m), where n is the number of vertices, and m and r respectively upper bound the number of hyperedges and the rank of the hypergraph at any time. 
We also extend our approach to the parallel batch-dynamic setting, where a batch of any k hyperedge insertions or deletions can be processed with O(kr² log ³ m) amortized work and O(log ² m) depth. This constitutes the first spectral-based sparsification algorithm in this setting.

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Sebastian Forster, Gramoz Goranci, and Ali Momeni. Fully Dynamic Spectral Sparsification for Directed Hypergraphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.38

Author Details

Sebastian Forster
  • Department of Computer Science, University of Salzburg, Austria
Gramoz Goranci
  • Faculty of Computer Science, University of Vienna, Austria
Ali Momeni
  • Faculty of Computer Science, UniVie Doctoral School Computer Science DoCS, University of Vienna, Austria

Funding

  • Forster, Sebastian: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 947702).

References

  1. Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, and Richard Peng. On fully dynamic graph sparsifiers. In 57th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 335-344, 2016. URL: https://doi.org/10.1109/FOCS.2016.44.
  2. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Spectral sparsification in dynamic graph streams. In Prasad Raghavendra, Sofya Raskhodnikova, Klaus Jansen, and José D. P. Rolim, editors, 16th Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX), volume 8096 of Lecture Notes in Computer Science, pages 1-10. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_1.
  3. Kyriakos Axiotis, Aleksander Madry, and Adrian Vladu. Faster sparse minimum cost flow by electrical flow localization. In 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 528-539, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00059.
  4. Nikhil Bansal, Ola Svensson, and Luca Trevisan. New notions and constructions of sparsification for graphs and hypergraphs. In 60th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 910-928, 2019. URL: https://doi.org/10.1109/FOCS.2019.00059.
  5. Joshua Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM Journal on Computing, 41(6):1704-1721, 2012. URL: https://doi.org/10.1137/090772873.
  6. Joshua D. Batson, Daniel A. Spielman, Nikhil Srivastava, and Shang-Hua Teng. Spectral sparsification of graphs: theory and algorithms. Commun. ACM, 56(8):87-94, 2013. URL: https://doi.org/10.1145/2492007.2492029.
  7. András A. Benczúr and David R. Karger. Approximating s-t minimum cuts in O(n²) time. In 28th Annual ACM Symposium on Theory of Computing (STOC), pages 47-55, 1996. URL: https://doi.org/10.1145/237814.237827.
  8. Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg, Danupon Nanongkai, Thatchaphol Saranurak, Aaron Sidford, and He Sun. Fully-dynamic graph sparsifiers against an adaptive adversary. In 49th International Colloquium on Automata, Languages and Programming (ICALP), volume 229, pages 20:1-20:20, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.20.
  9. Erik G. Boman, Bruce Hendrickson, and Stephen A. Vavasis. Solving elliptic finite element systems in near-linear time with support preconditioners. SIAM J. Numer. Anal., 46(6):3264-3284, 2008. URL: https://doi.org/10.1137/040611781.
  10. Yuri Boykov and Olga Veksler. Graph cuts in vision and graphics: Theories and applications. In Handbook of Mathematical Models in Computer Vision, pages 79-96. 2006. URL: https://doi.org/10.1007/0-387-28831-7_5.
  11. Li Chen, Gramoz Goranci, Monika Henzinger, Richard Peng, and Thatchaphol Saranurak. Fast dynamic cuts, distances and effective resistances via vertex sparsifiers. In 61st IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 1135-1146, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00109.
  12. Herman Chernoff. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. The Annals of Mathematical Statistics, 23(4):493-507, 1952. URL: http://www.jstor.org/stable/2236576.
  13. Michael B. Cohen, Cameron Musco, and Jakub Pachocki. Online row sampling. Theory Comput., 16:1-25, 2020. URL: https://doi.org/10.4086/TOC.2020.V016A015.
  14. Samuel I. Daitch and Daniel A. Spielman. Faster approximate lossy generalized flow via interior point algorithms. In 40th Annual ACM Symposium on Theory of Computing (STOC), pages 451-460, 2008. URL: https://doi.org/10.1145/1374376.1374441.
  15. Sally Dong, Yu Gao, Gramoz Goranci, Yin Tat Lee, Richard Peng, Sushant Sachdeva, and Guanghao Ye. Nested dissection meets ipms: Planar min-cost flow in nearly-linear time. In 33rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 124-153, 2022. URL: https://doi.org/10.1137/1.9781611977073.7.
  16. David Durfee, Yu Gao, Gramoz Goranci, and Richard Peng. Fully dynamic spectral vertex sparsifiers and applications. In 51st Annual ACM Symposium on Theory of Computing (STOC), pages 914-925, 2019. URL: https://doi.org/10.1145/3313276.3316379.
  17. Zhuo Feng. Similarity-aware spectral sparsification by edge filtering. In Proceedings of the 55th Annual Design Automation Conference, pages 1-6, 2018. URL: https://doi.org/10.1145/3195970.3196114.
  18. Sebastian Forster, Gramoz Goranci, Yang P. Liu, Richard Peng, Xiaorui Sun, and Mingquan Ye. Minor sparsifiers and the distributed laplacian paradigm. In 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 989-999, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00099.
  19. Yu Gao, Yang P. Liu, and Richard Peng. Fully dynamic electrical flows: Sparse maxflow faster than goldberg-rao. In 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 516-527, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00058.
  20. Mohsen Ghaffari and Jaehyun Koo. Parallel batch-dynamic algorithms for spanners, and extensions. In 37th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 299-313. ACM, 2025. URL: https://doi.org/10.1145/3694906.3743322.
  21. Gramoz Goranci, Monika Henzinger, and Pan Peng. Dynamic effective resistances and approximate schur complement on separable graphs. In 26th European Symposium on Algorithms (ESA), volume 112, pages 40:1-40:15, 2018. URL: https://doi.org/10.4230/LIPIcs.ESA.2018.40.
  22. Gramoz Goranci and Ali Momeni. Fully dynamic spectral sparsification of hypergraphs. CoRR, abs/2502.01421, 2025. URL: https://doi.org/10.48550/arXiv.2502.01421.
  23. Arun Jambulapati, Yang P. Liu, and Aaron Sidford. Chaining, group leverage score overestimates, and fast spectral hypergraph sparsification. In 55th Annual ACM Symposium on Theory of Computing (STOC), pages 196-206, 2023. URL: https://doi.org/10.1145/3564246.3585136.
  24. Michael Kapralov, Robert Krauthgamer, Jakab Tardos, and Yuichi Yoshida. Spectral hypergraph sparsifiers of nearly linear size. In 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 1159-1170, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00114.
  25. Michael Kapralov, Robert Krauthgamer, Jakab Tardos, and Yuichi Yoshida. Towards tight bounds for spectral sparsification of hypergraphs. In 53rd Annual ACM Symposium on Theory of Computing (STOC), pages 598-611, 2021. URL: https://doi.org/10.1145/3406325.3451061.
  26. Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. SIAM J. Comput., 46(1):456-477, 2017. URL: https://doi.org/10.1137/141002281.
  27. Michael Kapralov, Aida Mousavifar, Cameron Musco, Christopher Musco, Navid Nouri, Aaron Sidford, and Jakab Tardos. Fast and space efficient spectral sparsification in dynamic streams. In Shuchi Chawla, editor, ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1814-1833. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.111.
  28. Michael Kapralov, Navid Nouri, Aaron Sidford, and Jakab Tardos. Dynamic streaming spectral sparsification in nearly linear time and space. CoRR, abs/1903.12150, 2019. URL: https://arxiv.org/abs/1903.12150.
  29. Michael Kapralov and Rina Panigrahy. Spectral sparsification via random spanners. In Shafi Goldwasser, editor, Innovations in Theoretical Computer Science (ITCS), pages 393-398. ACM, 2012. URL: https://doi.org/10.1145/2090236.2090267.
  30. Jonathan A. Kelner and Alex Levin. Spectral sparsification in the semi-streaming setting. Theory Comput. Syst., 53(2):243-262, 2013. URL: https://doi.org/10.1007/S00224-012-9396-1.
  31. Jonathan A. Kelner and Aleksander Madry. Faster generation of random spanning trees. In 50th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 13-21, 2009. URL: https://doi.org/10.1109/FOCS.2009.75.
  32. Sanjeev Khanna, Huan Li, and Aaron Putterman. Near-optimal linear sketches and fully-dynamic algorithms for hypergraph spectral sparsification. In Michal Koucký and Nikhil Bansal, editors, 57th Annual ACM Symposium on Theory of Computing (STOC), pages 1190-1200. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718239.
  33. Sanjeev Khanna, Aaron (Louie) Putterman, and Madhu Sudan. Almost-tight bounds on preserving cuts in classes of submodular hypergraphs. In Karl Bringmann, Martin Grohe, Gabriele Puppis, and Ola Svensson, editors, 51st International Colloquium on Automata, Languages and Programming (ICALP), volume 297 of LIPIcs, pages 98:1-98:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.98.
  34. Pushmeet Kohli and Philip H. S. Torr. Dynamic graph cuts and their applications in computer vision. In Computer Vision: Detection, Recognition and Reconstruction, volume 285 of Studies in Computational Intelligence, pages 51-108. 2010. URL: https://doi.org/10.1007/978-3-642-12848-6_3.
  35. Ioannis Koutis, Alex Levin, and Richard Peng. Faster spectral sparsification and numerical algorithms for SDD matrices. ACM Trans. Algorithms, 12(2):17:1-17:16, 2016. URL: https://doi.org/10.1145/2743021.
  36. Ioannis Koutis, Gary L. Miller, and David Tolliver. Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing. Comput. Vis. Image Underst., 115(12):1638-1646, 2011. URL: https://doi.org/10.1016/J.CVIU.2011.05.013.
  37. Ioannis Koutis and Shen Chen Xu. Simple parallel and distributed algorithms for spectral graph sparsification. ACM Trans. Parallel Comput., 3(2):14:1-14:14, 2016. URL: https://doi.org/10.1145/2948062.
  38. James R. Lee. Spectral hypergraph sparsification via chaining. In 55th Annual ACM Symposium on Theory of Computing (STOC), pages 207-218, 2023. URL: https://doi.org/10.1145/3564246.3585165.
  39. James R. Lee, Shayan Oveis Gharan, and Luca Trevisan. Multiway spectral partitioning and higher-order cheeger inequalities. Journal of the ACM, 61(6):1-30, 2014. URL: https://doi.org/10.1145/2665063.
  40. Yin Tat Lee and He Sun. An sdp-based algorithm for linear-sized spectral sparsification. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, 49th Annual ACM Symposium on Theory of Computing (STOC), pages 678-687. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055477.
  41. Huan Li and Aaron Schild. Spectral subspace sparsification. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 385-396. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00044.
  42. Aleksander Madry. Fast approximation algorithms for cut-based problems in undirected graphs. In 51st IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 245-254, 2010. URL: https://doi.org/10.1109/FOCS.2010.30.
  43. Michael Mitzenmacher and Eli Upfal. Probability and computing: Randomization and probabilistic techniques in algorithms and data analysis. Cambridge university press, 2017. Google Scholar
  44. Kazusato Oko, Shinsaku Sakaue, and Shin-ichi Tanigawa. Nearly tight spectral sparsification of directed hypergraphs. In 50th International Colloquium on Automata, Languages and Programming (ICALP), pages 94:1-94:19, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.94.
  45. Richard Peng. Approximate undirected maximum flows in O(m polylog(n)) time. In 27th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1862-1867, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH130.
  46. Richard Peng, He Sun, and Luca Zanetti. Partitioning well-clustered graphs: Spectral clustering works! SIAM Journal on Computing, 46(2):710-743, 2017. URL: https://doi.org/10.1137/15m1047209.
  47. Akbar Rafiey and Yuichi Yoshida. Sparsification of decomposable submodular functions. In 36th Conference on Artificial Intelligence (AAAI), pages 10336-10344. AAAI Press, 2022. URL: https://doi.org/10.1609/AAAI.V36I9.21275.
  48. Jonah Sherman. Nearly maximum flows in nearly linear time. In 54th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 263-269, 2013. URL: https://doi.org/10.1109/FOCS.2013.36.
  49. Tasuku Soma, Kam Chuen Tung, and Yuichi Yoshida. Online algorithms for spectral hypergraph sparsification. In Jens Vygen and Jaroslaw Byrka, editors, 25th Conference on Integer Programming and Combinatorial Optimization (IPCO), volume 14679 of Lecture Notes in Computer Science, pages 405-417. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-59835-7_30.
  50. Tasuku Soma and Yuichi Yoshida. Spectral sparsification of hypergraphs. In Timothy M. Chan, editor, 30th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2570-2581. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.159.
  51. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. In 40th Annual ACM Symposium on Theory of Computing (STOC), 2008. URL: https://doi.org/10.1145/1374376.1374456.
  52. Daniel A. Spielman and Shang-Hua Teng. Spectral sparsification of graphs. SIAM J. Comput., 40(4):981-1025, 2011. URL: https://doi.org/10.1137/08074489X.
  53. Daniel A. Spielman and Shang-Hua Teng. Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM J. Matrix Anal. Appl., 35(3):835-885, 2014. URL: https://doi.org/10.1137/090771430.
  54. Jan van den Brand, Yu Gao, Arun Jambulapati, Yin Tat Lee, Yang P. Liu, Richard Peng, and Aaron Sidford. Faster maxflow via improved dynamic spectral vertex sparsifiers. In 54th Annual ACM Symposium on Theory of Computing (STOC), pages 543-556, 2022. URL: https://doi.org/10.1145/3519935.3520068.
  55. Yibin Zhao. Fully dynamic spectral and cut sparsifiers for directed graphs. CoRR, 2025. URL: https://doi.org/10.48550/arXiv.2507.19632.
  56. Chunjiang Zhu, Qinqing Liu, and Jinbo Bi. Spectral vertex sparsifiers and pair-wise spanners over distributed graphs. In 38th International Conference on Machine Learning (ICML), volume 139, pages 12890-12900, 2021. URL: https://proceedings.mlr.press/v139/zhu21c.html.
  57. Zeyuan Allen Zhu, Zhenyu Liao, and Lorenzo Orecchia. Spectral sparsification and regret minimization beyond matrix multiplicative updates. In Rocco A. Servedio and Ronitt Rubinfeld, editors, 47th Annual ACM Symposium on Theory of Computing (STOC), pages 237-245. ACM, 2015. URL: https://doi.org/10.1145/2746539.2746610.
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