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High-Accuracy Multicommodity Flows via Iterative Refinement

Authors Li Chen, Mingquan Ye

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Author Details

Li Chen
  • Carnegie Mellon University, Pittsburgh, PA, USA
Mingquan Ye
  • University of Illinois at Chicago, IL, USA

Funding

  • Chen, Li: Supported by NSF grants CCF-2106444 and CCF-2330255.
  • Ye, Mingquan: Supported by NSF grant CCF-2240024.

Acknowledgements

We thank Richard Peng and Xiaorui Sun for helpful discussions.

Cite As Get BibTex

Li Chen and Mingquan Ye. High-Accuracy Multicommodity Flows via Iterative Refinement. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.45

Abstract

The multicommodity flow problem is a classic problem in network flow and combinatorial optimization, with applications in transportation, communication, logistics, and supply chain management, etc. Existing algorithms often focus on low-accuracy approximate solutions, while high-accuracy algorithms typically rely on general linear program solvers. In this paper, we present efficient high-accuracy algorithms for a broad family of multicommodity flow problems on undirected graphs, demonstrating improved running times compared to general linear program solvers. Our main result shows that we can solve the 𝓁_{q, p}-norm multicommodity flow problem to a (1 + ε) approximation in time O_{q, p}(m^{1+o(1)} k² log(1/ε)), where k is the number of commodities, and O_{q, p}(⋅) hides constants depending only on q or p. As q and p approach to 1 and ∞ respectively, 𝓁_{q, p}-norm flow tends to maximum concurrent flow.
We introduce the first iterative refinement framework for 𝓁_{q, p}-norm minimization problems, which reduces the problem to solving a series of decomposable residual problems. In the case of k-commodity flow, each residual problem can be decomposed into k single commodity convex flow problems, each of which can be solved in almost-linear time. As many classical variants of multicommodity flows were shown to be complete for linear programs in the high-accuracy regime [Ding-Kyng-Zhang, ICALP'22], our result provides new directions for studying more efficient high-accuracy multicommodity flow algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Continuous optimization
Keywords
  • High-accuracy multicommodity flow
  • Iterative refinement framework
  • Convex flow solver

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References

  1. Deeksha Adil, Brian Bullins, Rasmus Kyng, and Sushant Sachdeva. Almost-linear-time weighted 𝓁_p-norm solvers in slightly dense graphs via sparsification. arXiv preprint, 2021. URL: https://arxiv.org/abs/2102.06977.
  2. Deeksha Adil, Rasmus Kyng, Richard Peng, and Sushant Sachdeva. Iterative refinement for 𝓁_p-norm regression. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1405-1424, 2019. Google Scholar
  3. Deeksha Adil, Rasmus Kyng, Richard Peng, and Sushant Sachdeva. Fast algorithms for 𝓁_p-regression. arXiv preprint, 2022. URL: https://arxiv.org/abs/2211.03963.
  4. Deeksha Adil, Richard Peng, and Sushant Sachdeva. Fast, provably convergent IRLS algorithm for p-norm linear regression. Advances in Neural Information Processing Systems (NeurIPS), 32, 2019. Google Scholar
  5. Deeksha Adil and Sushant Sachdeva. Faster p-norm minimizing flows, via smoothed q-norm problems. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 892-910, 2020. Google Scholar
  6. Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network flows: theory, algorithms, and applications. Prentice-Hall, Inc., USA, 1993. Google Scholar
  7. Kyriakos Axiotis, Aleksander Mądry, and Adrian Vladu. Circulation control for faster minimum cost flow in unit-capacity graphs. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 93-104, 2020. Google Scholar
  8. Kyriakos Axiotis, Aleksander Mądry, and Adrian Vladu. Faster sparse minimum cost flow by electrical flow localization. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 528-539, 2022. Google Scholar
  9. Cynthia Barnhart, Niranjan Krishnan, and Pamela H Vance. Multicommodity flow problems. Encyclopedia of Optimization, 14:2354-2362, 2009. Google Scholar
  10. Aaron Bernstein, Maximilian Probst Gutenberg, and Thatchaphol Saranurak. Deterministic decremental SSSP and approximate min-cost flow in almost-linear time. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 1000-1008, 2022. Google Scholar
  11. Daniel Bienstock and Garud Iyengar. Solving fractional packing problems in Õ(1/ε) iterations. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing (STOC), pages 146-155, 2004. Google Scholar
  12. Jan van den Brand, Yin-Tat Lee, Danupon Nanongkai, Richard Peng, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, and Di Wang. Bipartite matching in nearly-linear time on moderately dense graphs. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 919-930, 2020. Google Scholar
  13. Jan van den Brand and Daniel Zhang. Faster high accuracy multi-commodity flow from single-commodity techniques. arXiv preprint, 2023. URL: https://arxiv.org/abs/2304.12992.
  14. Sébastien Bubeck, Michael B Cohen, Yin Tat Lee, and Yuanzhi Li. An homotopy method for lp regression provably beyond self-concordance and in input-sparsity time. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 1130-1137, 2018. Google Scholar
  15. 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 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 612-623, 2022. Google Scholar
  16. Li Chen, Richard Peng, and Di Wang. 2-norm flow diffusion in near-linear time. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 540-549, 2022. Google Scholar
  17. Paul Christiano, Jonathan A Kelner, Aleksander Madry, Daniel A Spielman, and Shang-Hua Teng. Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs. In Proceedings of the forty-third annual ACM symposium on Theory of computing (STOC), pages 273-282, 2011. Google Scholar
  18. Kenneth Clarkson, Ruosong Wang, and David Woodruff. Dimensionality reduction for tukey regression. In International Conference on Machine Learning (ICML), pages 1262-1271, 2019. Google Scholar
  19. Kenneth L Clarkson, Petros Drineas, Malik Magdon-Ismail, Michael W Mahoney, Xiangrui Meng, and David P Woodruff. The fast cauchy transform and faster robust linear regression. SIAM Journal on Computing, 45(3):763-810, 2016. Google Scholar
  20. Kenneth L Clarkson and David P Woodruff. Low-rank approximation and regression in input sparsity time. Journal of the ACM (JACM), 63(6):1-45, 2017. Google Scholar
  21. Michael B Cohen, Yin Tat Lee, and Zhao Song. Solving linear programs in the current matrix multiplication time. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 938-942, 2019. Google Scholar
  22. Michael B Cohen, Yin Tat Lee, and Zhao Song. Solving linear programs in the current matrix multiplication time. Journal of the ACM (JACM), 68(1):1-39, 2021. Google Scholar
  23. Michael B Cohen, Aleksander Mądry, Piotr Sankowski, and Adrian Vladu. Negative-weight shortest paths and unit capacity minimum cost flow in Õ(m^10/7 log w) time. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 752-771, 2017. Google Scholar
  24. Samuel I Daitch and Daniel A Spielman. Faster approximate lossy generalized flow via interior point algorithms. In Proceedings of the fortieth annual ACM symposium on Theory of computing (STOC), pages 451-460, 2008. Google Scholar
  25. Anirban Dasgupta, Petros Drineas, Boulos Harb, Ravi Kumar, and Michael W Mahoney. Sampling algorithms and coresets for 𝓁_p regression. SIAM Journal on Computing, 38(5):2060-2078, 2009. Google Scholar
  26. Ming Ding, Rasmus Kyng, and Peng Zhang. Two-commodity flow is equivalent to linear programming under nearly-linear time reductions. In 49th International Colloquium on Automata, Languages, and Programming (ICALP), pages 54:1-54:19, 2022. Google Scholar
  27. Lisa K Fleischer. Approximating fractional multicommodity flow independent of the number of commodities. SIAM Journal on Discrete Mathematics, 13(4):505-520, 2000. Google Scholar
  28. Sebastian Forster, Gramoz Goranci, Yang P Liu, Richard Peng, Xiaorui Sun, and Mingquan Ye. Minor sparsifiers and the distributed laplacian paradigm. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 989-999, 2022. Google Scholar
  29. Yu Gao, Yang P Liu, and Richard Peng. Fully dynamic electrical flows: Sparse maxflow faster than goldberg-rao. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 516-527, 2022. Google Scholar
  30. Naveen Garg and Jochen Könemann. Faster and simpler algorithms for multicommodity flow and other fractional packing problems. SIAM Journal on Computing, 37(2):630-652, 2007. Google Scholar
  31. Mehrdad Ghadiri, Richard Peng, and Santosh S Vempala. Faster p-norm regression using sparsity. arXiv preprint arXiv:2109.11537, 2021. Google Scholar
  32. Andrew V Goldberg. A natural randomization strategy for multicommodity flow and related algorithms. Information Processing Letters, 42(5):249-256, 1992. Google Scholar
  33. Michael D Grigoriadis and Leonid G Khachiyan. Fast approximation schemes for convex programs with many blocks and coupling constraints. SIAM Journal on Optimization, 4(1):86-107, 1994. Google Scholar
  34. Michael D Grigoriadis and Leonid G Khachiyan. Approximate minimum-cost multicommodity flows in Õ(ε^-2knm) time. Mathematical Programming, 75(3):477-482, 1996. Google Scholar
  35. Alon Itai. Two-commodity flow. Journal of the ACM (JACM), 25(4):596-611, 1978. Google Scholar
  36. Sanjiv Kapoor and Pravin M Vaidya. Speeding up karmarkar’s algorithm for multicommodity flows. Mathematical programming, 73:111-127, 1996. Google Scholar
  37. George Karakostas. Faster approximation schemes for fractional multicommodity flow problems. In Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 166-173, 2002. Google Scholar
  38. David Karger and Serge Plotkin. Adding multiple cost constraints to combinatorial optimization problems, with applications to multicommodity flows. In Proceedings of the twenty-seventh annual ACM symposium on Theory of computing (STOC), pages 18-25, 1995. Google Scholar
  39. Narendra Karmarkar. A new polynomial-time algorithm for linear programming. In Proceedings of the sixteenth annual ACM symposium on Theory of computing (STOC), pages 302-311, 1984. Google Scholar
  40. Tarun Kathuria, Yang P. Liu, and Aaron Sidford. Unit capacity maxflow in almost O(m^4/3) time. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 119-130, 2020. Google Scholar
  41. 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 the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms (SODA), pages 217-226, 2014. Google Scholar
  42. Jonathan A Kelner, Gary L Miller, and Richard Peng. Faster approximate multicommodity flow using quadratically coupled flows. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing (STOC), pages 1-18, 2012. Google Scholar
  43. Jeff L Kennington. A survey of linear cost multicommodity network flows. Operations Research, 26(2):209-236, 1978. Google Scholar
  44. Leonid G Khachiyan. Polynomial algorithms in linear programming. USSR Computational Mathematics and Mathematical Physics, 20(1):53-72, 1980. Google Scholar
  45. Philip Klein, Serge Plotkin, Clifford Stein, and Eva Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466-487, 1994. Google Scholar
  46. Rasmus Kyng, Richard Peng, Sushant Sachdeva, and Di Wang. Flows in almost linear time via adaptive preconditioning. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 902-913, 2019. Google Scholar
  47. Yin Tat Lee and Aaron Sidford. Solving linear programs with sqrt (rank) linear system solves. arXiv preprint, 2019. URL: https://arxiv.org/abs/1910.08033.
  48. T. Leighton, F. Makedon, S. Plotkin, C. Stein, E. Tardos, and S. Tragoudas. Fast approximation algorithms for multicommodity flow problems. Journal of Computer and System Sciences, 50(2):228-243, 1995. Google Scholar
  49. Tom Leighton, Clifford Stein, Fillia Makedon, Éva Tardos, Serge Plotkin, and Spyros Tragoudas. Fast approximation algorithms for multicommodity flow problems. In Proceedings of the twenty-third annual ACM symposium on Theory of Computing (STOC), pages 101-111, 1991. Google Scholar
  50. Jason Li. Faster parallel algorithm for approximate shortest path. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 308-321, 2020. Google Scholar
  51. Aleksander Madry. Faster approximation schemes for fractional multicommodity flow problems via dynamic graph algorithms. In Proceedings of the forty-second ACM symposium on Theory of computing (STOC), pages 121-130, 2010. Google Scholar
  52. 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 (FOCS), pages 253-262, 2013. Google Scholar
  53. Aleksander Madry. Computing maximum flow with augmenting electrical flows. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 593-602, 2016. Google Scholar
  54. Xiangrui Meng and Michael Mahoney. Robust regression on mapreduce. In International Conference on Machine Learning (ICML), pages 888-896, 2013. Google Scholar
  55. Arkadi Nemirovski. Interior point polynomial time methods in convex programming. Lecture notes, 42(16):3215-3224, 2004. Google Scholar
  56. Yurii Nesterov. Fast gradient methods for network flow problems. In Proceeding of the 20th International Symposium of Mathematical Programming (ISMP), 2009. Google Scholar
  57. Adamou Ouorou, Philippe Mahey, and J-Ph Vial. A survey of algorithms for convex multicommodity flow problems. Management science, 46(1):126-147, 2000. Google Scholar
  58. Richard Peng. Approximate undirected maximum flows in O(m polylog(n)) time. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms (SODA), pages 1862-1867, 2016. Google Scholar
  59. Richard Peng and Santosh Vempala. Solving sparse linear systems faster than matrix multiplication. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 504-521, 2021. Google Scholar
  60. Serge A Plotkin, David B Shmoys, and Éva Tardos. Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research, 20(2):257-301, 1995. Google Scholar
  61. Tomasz Radzik. Fast deterministic approximation for the multicommodity flow problem. Mathematical Programming, 78:43-58, 1996. Google Scholar
  62. James Renegar. A polynomial-time algorithm, based on newton’s method, for linear programming. Mathematical programming, 40(1-3):59-93, 1988. Google Scholar
  63. Bruce Rothschild and Andrew Whinston. Feasibility of two commodity network flows. Operations Research, 14(6):1121-1129, 1966. Google Scholar
  64. Farhad Shahrokhi and David W Matula. The maximum concurrent flow problem. Journal of the ACM (JACM), 37(2):318-334, 1990. Google Scholar
  65. Jonah Sherman. Nearly maximum flows in nearly linear time. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pages 263-269, 2013. Google Scholar
  66. Jonah Sherman. Area-convexity, 𝓁_∞ regularization, and undirected multicommodity flow. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 452-460, 2017. Google Scholar
  67. David B Shmoys. Cut problems and their application to divide-and-conquer. Approximation algorithms for NP-hard problems, pages 192-235, 1997. Google Scholar
  68. D. Spielman and S. Teng. Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM Journal on Matrix Analysis and Applications, 35(3):835-885, 2014. Available at http://arxiv.org/abs/cs/0607105. Google Scholar
  69. 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 Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 543-556, 2022. Google Scholar
  70. Jan van den Brand, Yin Tat Lee, Yang P Liu, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, and Di Wang. Minimum cost flows, MDPs, and 𝓁₁-regression in nearly linear time for dense instances. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 859-869, 2021. Google Scholar
  71. I-Lin Wang. Multicommodity network flows: A survey, part i: Applications and formulations. International Journal of Operations Research, 15(4):145-153, 2018. Google Scholar
  72. David Woodruff and Qin Zhang. Subspace embeddings and 𝓁_p-regression using exponential random variables. In Conference on Learning Theory (COLT), pages 546-567, 2013. Google Scholar
  73. Neal E. Young. Randomized rounding without solving the linear program. In Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 170-178, 1995. Google Scholar
  74. Goran Zuzic, Gramoz Goranci, Mingquan Ye, Bernhard Haeupler, and Xiaorui Sun. Universally-optimal distributed shortest paths and transshipment via graph-based 𝓁₁-oblivious routing. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2549-2579, 2022. Google Scholar
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