Document Open Access Logo

Length-Constrained Directed Expander Decomposition and Length-Constrained Vertex-Capacitated Flow Shortcuts

Authors Bernhard Haeupler , Yaowei Long , Thatchaphol Saranurak , Shengzhe Wang

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.ESA.2025.107.pdf
  • Filesize: 0.86 MB
  • 17 pages
HTML5 Logo HTML (experimental)
LIPIcs.ESA.2025.107.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Length-Constrained Expander
  • Expander Decomposition
  • Shortcut

Metrics

Abstract

We show the existence of length-constrained expander decomposition in directed graphs and undirected vertex-capacitated graphs. Previously, its existence was shown only in undirected edge-capacitated graphs [Bernhard Haeupler et al., 2022; Haeupler et al., 2024]. Along the way, we prove the multi-commodity maxflow-mincut theorems for length-constrained expansion in both directed and undirected vertex-capacitated graphs. Based on our decomposition, we build a length-constrained flow shortcut for undirected vertex-capacitated graphs, which roughly speaking is a set of edges and vertices added to the graph so that every multi-commodity flow demand can be routed with approximately the same vertex-congestion and length, but all flow paths only contain few edges. This generalizes the shortcut for undirected edge-capacitated graphs from [Bernhard Haeupler et al., 2024]. Length-constrained expander decomposition and flow shortcuts have been crucial in the recent algorithms in undirected edge-capacitated graphs [Bernhard Haeupler et al., 2024; Haeupler et al., 2024]. Our work thus serves as a foundation to generalize these concepts to directed and vertex-capacitated graphs.

Cite As Get BibTex

Bernhard Haeupler, Yaowei Long, Thatchaphol Saranurak, and Shengzhe Wang. Length-Constrained Directed Expander Decomposition and Length-Constrained Vertex-Capacitated Flow Shortcuts. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 107:1-107:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.107

Author Details

Bernhard Haeupler
  • INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria
  • ETH Zürich, Switzerland
Yaowei Long
  • University of Michigan, Ann Arbor, MI, USA
Thatchaphol Saranurak
  • University of Michigan, Ann Arbor, MI, USA
Shengzhe Wang
  • ETH Zürich, Switzerland

Funding

Part of this work was done while at INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria. This work was partially funded from the Ministry of Education and Science of Bulgaria (support for INSAIT, part of the Bulgarian National Roadmap for Research Infrastructure).
  • Haeupler, Bernhard: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERC grant agreement 949272).
  • Saranurak, Thatchaphol: Supported by NSF Grant CCF-2238138.
  • Wang, Shengzhe: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC grant agreement No 949272 and No 101171685).

References

  1. Daniel Agassy, Dani Dorfman, and Haim Kaplan. Expander decomposition with fewer inter-cluster edges using a spectral cut player. In Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP), pages 9:1-9:20, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.9.
  2. Aaron Bernstein, Joakim Blikstad, Thatchaphol Saranurak, and Ta-Wei Tu. Maximum flow by augmenting paths in n^2+o(1) time, 2024. URL: https://doi.org/10.48550/arXiv.2406.03648.
  3. Aaron Bernstein, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Deterministic decremental reachability, scc, and shortest paths via directed expanders and congestion balancing. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 1123-1134, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00108.
  4. Aaron Bernstein, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Deterministic decremental sssp and approximate min-cost flow in almost-linear time. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 1000-1008, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00100.
  5. 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 Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 20:1-20:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.20.
  6. Greg Bodwin and Gary Hoppenworth. Folklore sampling is optimal for exact hopsets: Confirming the √n barrier. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 701-720. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00046.
  7. Y. Chang and T. Saranurak. Deterministic distributed expander decomposition and routing with applications in distributed derandomization. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 377-388, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00043.
  8. Yi-Jun Chang and Thatchaphol Saranurak. Improved distributed expander decomposition and nearly optimal triangle enumeration. In Proceedings of the International Symposium on Principles of Distributed Computing (PODC), pages 66-73, 2019. URL: https://doi.org/10.1145/3293611.3331618.
  9. Chandra Chekuri and Alina Ene. The all-or-nothing flow problem in directed graphs with symmetric demand pairs. Mathematical Programming, 154:249-272, 2015. URL: https://doi.org/10.1007/S10107-014-0856-Z.
  10. Chandra Chekuri, Sanjeev Khanna, and F Bruce Shepherd. Multicommodity flow, well-linked terminals, and routing problems. In Proceedings of the Symposium on Theory of Computing (STOC), pages 183-192, 2005. URL: https://doi.org/10.1145/1060590.1060618.
  11. Daoyuan Chen, Simon Meierhans, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Parallel and distributed expander decomposition: Simple, fast, and near-optimal. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1705-1719, 2025. URL: https://doi.org/10.1137/1.9781611978322.53.
  12. 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 Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 612-623, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00064.
  13. 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 Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 1158-1167, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00111.
  14. Julia Chuzhoy and Sanjeev Khanna. A new algorithm for decremental single-source shortest paths with applications to vertex-capacitated flow and cut problems. In Proceedings of the Symposium on Theory of Computing (STOC), pages 389-400, 2019. URL: https://doi.org/10.1145/3313276.3316320.
  15. Julia Chuzhoy and Sanjeev Khanna. Maximum bipartite matching in n^2+o(1) time via a combinatorial algorithm. In Proceedings of the Symposium on Theory of Computing (STOC), pages 83-94, 2024. URL: https://doi.org/10.1145/3618260.3649725.
  16. Antoine El-Hayek, Monika Henzinger, and Jason Li. Fully dynamic approximate minimum cut in subpolynomial time per operation. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 750-784, 2025. URL: https://doi.org/10.1137/1.9781611978322.22.
  17. Oded Goldreich and Dana Ron. A sublinear bipartiteness tester for bounded degree graphs. In Proceedings of the Symposium on Theory of Computing (STOC), pages 289-298, 1998. URL: https://doi.org/10.1145/276698.276767.
  18. Gramoz Goranci, Harald Räcke, Thatchaphol Saranurak, and Zihan Tan. The expander hierarchy and its applications to dynamic graph algorithms. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2212-2228, 2021. URL: https://doi.org/10.1137/1.9781611976465.132.
  19. Lars Gottesbüren, Nikos Parotsidis, and Maximilian Probst Gutenberg. Practical expander decomposition. In Proceedings of the European Symposium on Algorithms (ESA), pages 61:1-61:17, 2024. URL: https://doi.org/10.4230/LIPICS.ESA.2024.61.
  20. Bernhard Haeupler, D. Ellis Hershkowitz, Jason Li, Antti Roeyskoe, and Thatchaphol Saranurak. Low-step multi-commodity flow emulators. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 71-82. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649689.
  21. Bernhard Haeupler, D Ellis Hershkowitz, and Zihan Tan. New structures and algorithms for length-constrained expander decompositions. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 1634-1645, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00102.
  22. Bernhard Haeupler, Jonas Huebotter, and Mohsen Ghaffari. A cut-matching game for constant-hop expanders. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1651-1678, 2025. URL: https://doi.org/10.1137/1.9781611978322.51.
  23. Bernhard Haeupler, Yaowei Long, and Thatchaphol Saranurak. Dynamic deterministic constant-approximate distance oracles with n^ε worst-case update time. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 2033-2044, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00121.
  24. Bernhard Haeupler, Harald Räcke, and Mohsen Ghaffari. Hop-constrained expander decompositions, oblivious routing, and distributed universal optimality. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1325-1338. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520026.
  25. Bernhard Haeupler, David Wajc, and Goran Zuzic. Network coding gaps for completion times of multiple unicasts. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 494-505, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00053.
  26. William Hesse. Directed graphs requiring large numbers of shortcuts. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 665-669, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644216.
  27. Yiding Hua, Rasmus Kyng, Maximilian Probst Gutenberg, and Zihang Wu. Maintaining expander decompositions via sparse cuts. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 48-69, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH2.
  28. Shang-En Huang and Seth Pettie. Lower bounds on sparse spanners, emulators, and diameter-reducing shortcuts. SIAM J. Discret. Math., 35(3):2129-2144, 2021. URL: https://doi.org/10.1137/19M1306154.
  29. Yonggang Jiang, Merav Parter, and Asaf Petruschka. New oracles and labeling schemes for vertex cut queries, 2025. URL: https://doi.org/10.48550/arXiv.2501.13596.
  30. Wenyu Jin and Xiaorui Sun. Fully dynamic s-t edge connectivity in subpolynomial time (extended abstract). In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 861-872. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00088.
  31. Wenyu Jin, Xiaorui Sun, and Mikkel Thorup. Fully dynamic min-cut of superconstant size in subpolynomial time. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2999-3026, 2024. URL: https://doi.org/10.1137/1.9781611977912.107.
  32. Ravi Kannan, Santosh Vempala, and Adrian Vetta. On clusterings: Good, bad and spectral. Journal of the ACM (JACM), 51(3):497-515, 2004. URL: https://doi.org/10.1145/990308.990313.
  33. 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 Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 217-226, 2014. URL: https://doi.org/10.1137/1.9781611973402.16.
  34. Tom Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM (JACM), 46(6):787-832, 1999. URL: https://doi.org/10.1145/331524.331526.
  35. Jason Li. Deterministic mincut in almost-linear time. In Proceedings of the Symposium on Theory of Computing (STOC), pages 384-395, 2021. URL: https://doi.org/10.1145/3406325.3451114.
  36. Jason Li, Danupon Nanongkai, Debmalya Panigrahi, and Thatchaphol Saranurak. Near-linear time approximations for cut problems via fair cuts. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 240-275, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH10.
  37. Jason Li and Thatchaphol Saranurak. Deterministic weighted expander decomposition in almost-linear time, 2021. URL: https://doi.org/10.48550/arXiv.2106.01567.
  38. Yaowei Long, Seth Pettie, and Thatchaphol Saranurak. Connectivity labeling schemes for edge and vertex faults via expander hierarchies. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1-47, 2025. URL: https://doi.org/10.1137/1.9781611978322.1.
  39. Yaowei Long and Thatchaphol Saranurak. Near-optimal deterministic vertex-failure connectivity oracles. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 1002-1010, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00098.
  40. Chaitanya Nalam, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Deterministic k-vertex connectivity in k² max-flows, 2023. URL: https://doi.org/10.48550/arXiv.2308.04695.
  41. Danupon Nanongkai, Thatchaphol Saranurak, and Christian Wulff-Nilsen. Dynamic minimum spanning forest with subpolynomial worst-case update time. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 950-961, October 2017. URL: https://doi.org/10.1109/FOCS.2017.92.
  42. Harald Räcke, Chintan Shah, and Hanjo Täubig. Computing cut-based hierarchical decompositions in almost linear time. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 227-238, 2014. URL: https://doi.org/10.1137/1.9781611973402.17.
  43. Thatchaphol Saranurak and Di Wang. Expander decomposition and pruning: Faster, stronger, and simpler. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2616-2635, 2019. URL: https://doi.org/10.1137/1.9781611975482.162.
  44. Thatchaphol Saranurak and Sorrachai Yingchareonthawornchai. Deterministic small vertex connectivity in almost linear time. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 789-800, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00080.
  45. Jonah Sherman. Nearly maximum flows in nearly linear time. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 263-269, 2013. URL: https://doi.org/10.1109/FOCS.2013.36.
  46. Daniel A Spielman and Shang-Hua Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the Symposium on Theory of Computing (STOC), pages 81-90, 2004. URL: https://doi.org/10.1145/1007352.1007372.
  47. Aurelio L. Sulser and Maximilian Probst Gutenberg. A simple and near-optimal algorithm for directed expander decompositions. CoRR, abs/2403.04542, 2024. URL: https://doi.org/10.48550/arXiv.2403.04542.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact