High-Accuracy Multicommodity Flows via Iterative Refinement
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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) k^2 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 infinity 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.
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