2026-07-23T18:55:24Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20188 2024-07-02T07:52:50Z ddc:004 open_access

High-Accuracy Multicommodity Flows via Iterative Refinement Chen, Li Ye, Mingquan High-accuracy multicommodity flow Iterative refinement framework Convex flow solver 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Li Chen and Mingquan Ye 2024 Is Part Of LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2024.45 urn:nbn:de:0030-drops-201887 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.45 eng https://creativecommons.org/licenses/by/4.0/legalcode