Accelerated Approximate Optimization of Multi-commodity Flows on Directed Graphs

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Li Chen, Andrei Graur, Aaron Sidford

Published: 2025, Last Modified: 03 Oct 2025STOC 2025EveryoneRevisionsBibTeXCC BY-SA 4.0
Abstract: We provide m1+o(1)kє−1-time algorithms for computing multiplicative (1 − є)-approximate solutions to multi-commodity flow problems with k-commodities on m-edge directed graphs, including concurrent multi-commodity flow and maximum multi-commodity flow.To obtain our results, we provide new optimization tools of potential independent interest.First, we provide an improved optimization method for solving ℓq, p-regression problems to high accuracy. This method makes Õq, p(k) queries to a high accuracy convex minimization oracle, where Õq, p(·) hides factors depending only on q, p, or poly(logm), improving upon the Õq, p(k2) bound of [Chen-Ye, ICALP 2024].As a result, we obtain the first almost-linear time algorithm that solves ℓq, p flows on directed graphs to high accuracy.Second, we present optimization tools to reduce approximately solving composite ℓ1, ∞-regression problems to solving mo(1)є−1 instances of the composite ℓq, p-regression problem.The method builds upon recent advances in solving box-simplex games [Jambulapati-Tian, NeurIPS 2023] and the area convex regularizer introduced in [Sherman, STOC 2017] to obtain faster rates for constrained versions of the problem. Carefully combining these techniques yields our directed multi-commodity flow algorithm.