Low-Step Multi-commodity Flow Emulators
Authors: 
Bernhard Haeupler, 
D. Ellis Hershkowitz, Jason Li, Antti Roeyskoe, and Thatchaphol SaranurakAuthors Info & Claims
Bernhard Haeupler, D. Ellis Hershkowitz, Jason Li, Antti Roeyskoe, and Thatchaphol SaranurakAuthors Info & ClaimsJune 2024
Pages 71 - 82
Abstract
We introduce the concept of low-step multi-commodity flow emulators for any undirected, capacitated graph. At a high level, these emulators contain approximate multi-commodity flows whose paths contain a small number of edges, shattering the infamous flow decomposition barrier for multi-commodity flow.
We prove the existence of low-step multi-commodity flow emulators and develop efficient algorithms to compute them. We then apply them to solve constant-approximate k-commodity flow in O((m+k)1+є) time. To bypass the O(mk) flow decomposition barrier, we represent our output multi-commodity flow implicitly; prior to our work, even the existence of implicit constant-approximate multi-commodity flows of size o(mk) was unknown.
Our results generalize to the minimum cost setting, where each edge has an associated cost and the multi-commodity flow must satisfy a cost budget. Our algorithms are also parallel.
References
[1]
Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. 2022. Maximum Flow and Minimum-Cost Flow in Almost-Linear Time. 612–623. https://doi.org/10.1109/FOCS54457.2022.00064
[2]
Julia Chuzhoy and Sanjeev Khanna. 2009. Polynomial flow-cut gaps and hardness of directed cut problems. J. ACM, 56, 2 (2009), 6:1–6:28. https://doi.org/10.1145/1502793.1502795
[3]
Edith Cohen. 2000. Polylog-time and near-linear work approximation scheme for undirected shortest paths. J. ACM, 47, 1 (2000), 132–166. https://doi.org/10.1145/331605.331610
[4]
Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. 2022. Introduction to algorithms. MIT press.
[5]
Naveen Garg and Jochen Könemann. 2007. Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems. SIAM J. Comput., 37, 2 (2007), 630–652. https://doi.org/10.1137/S0097539704446232
[6]
Gramoz Goranci, Harald Räcke, Thatchaphol Saranurak, and Zihan Tan. 2021. The Expander Hierarchy and its Applications to Dynamic Graph Algorithms. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, Dániel Marx (Ed.). SIAM, 2212–2228. https://doi.org/10.1137/1.9781611976465.132
[7]
Bernhard Haeupler, D. Ellis Hershkowitz, and Thatchaphol Saranurak. 2023. Maximum Length-Constrained Flows and Disjoint Paths: Distributed, Deterministic, and Fast. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, Barna Saha and Rocco A. Servedio (Eds.). ACM, 1371–1383. https://doi.org/10.1145/3564246.3585202
[8]
Bernhard Haeupler, D. Ellis Hershkowitz, and Zihan Tan. 2023. Length-Constrained Expander Decompositions: Deeper Theory, Better Algorithms. Manuscript.
[9]
Bernhard Haeupler, D. Ellis Hershkowitz, and Zihan Tan. 2023. Parallel Greedy Spanners. CoRR, abs/2304.08892 (2023), https://doi.org/10.48550/ARXIV.2304.08892 arXiv:2304.08892.
[10]
Bernhard Haeupler, Jonas Hübotter, and Mohsen Ghaffari. 2022. A Cut-Matching Game for Constant-Hop Expanders. CoRR, abs/2211.11726 (2022), https://doi.org/10.48550/ARXIV.2211.11726 arXiv:2211.11726.
[11]
Bernhard Haeupler, Harald Räcke, and Mohsen Ghaffari. 2022. Hop-constrained expander decompositions, oblivious routing, and distributed universal optimality. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. 1325–1338.
[12]
Frank Thomson Leighton and Satish Rao. 1999. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM, 46, 6 (1999), 787–832. https://doi.org/10.1145/331524.331526
[13]
David Peleg. 2000. Distributed computing: a locality-sensitive approach. SIAM.
[14]
Harald Racke. 2002. Minimizing congestion in general networks. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings. 43–52.
[15]
Harald Räcke, Chintan Shah, and Hanjo Täubig. 2014. Computing cut-based hierarchical decompositions in almost linear time. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms. 227–238.
[16]
Jonah Sherman. 2017. Area-convexity, l_ ∞ regularization, and undirected multicommodity flow. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, Hamed Hatami, Pierre McKenzie, and Valerie King (Eds.). ACM, 452–460. https://doi.org/10.1145/3055399.3055501
[17]
Jan van den Brand and Daniel J. Zhang. 2023. Faster High Accuracy Multi-Commodity Flow from Single-Commodity Techniques. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023. IEEE, 493–502. https://doi.org/10.1109/FOCS57990.2023.00036
Index Terms
- Low-Step Multi-commodity Flow Emulators
Recommendations
Greedy distributed optimization of multi-commodity flows
PODC '07: Proceedings of the twenty-sixth annual ACM symposium on Principles of distributed computingThe multi-commodity flow problem is a classical combinatorial optimization problem that addresses a number of practically important issues of congestion and bandwidth management in connection-oriented network architectures.
We consider solutions for ...
Max-min fairness in multi-commodity flows
In this paper, we provide a study of Max-Min Fair (MMF) multi-commodity flows and focus on some of their applications to multi-commodity networks. We first present the theoretical background for the problem of MMF and recall its relations with ...
Greedy distributed optimization of multi-commodity flows
The multi-commodity flow problem is a classical combinatorial optimization problem that addresses a number of practically important issues of congestion and bandwidth management in connection-oriented network architectures. We consider solutions for ...
Comments
Information & Contributors
Information
Published In
June 2024
2049 pages
ISBN:9798400703836
DOI:10.1145/3618260
- General Chairs:
- Bojan Mohar,
- Igor Shinkar,
- Program Chair:
- Ryan O'Donnell
Copyright © 2024 Copyright is held by the owner/author(s). Publication rights licensed to ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
Sponsors
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 11 June 2024
Check for updates
Author Tags
Qualifiers
- Research-article
Funding Sources
- European Research Council
- NSF (National Science Foundation)
Conference
STOC '24
Sponsor:
Acceptance Rates
Overall Acceptance Rate 1,389 of 4,261 submissions, 33%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 61Total Downloads
- Downloads (Last 12 months)61
- Downloads (Last 6 weeks)61
Other Metrics
Citations
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in