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

Acknowledgements

We thank Richard Peng and Xiaorui Sun for helpful discussions.

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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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