Accelerated Approximate Optimization of Multi-Commodity Flows on Directed Graphs

2025-03-31

AI Summary

This paper addresses the approximate computation of multicommodity flows—including concurrent and maximum flows—on directed graphs. Methodologically, it introduces the first nearly linear-time ℓ_{q,p}-flow solver; innovatively reduces ℓ_{1,∞}-regression to m^{o(1)}/ε instances of ℓ_{q,p}-regression; and integrates area-convex regularization with the box-simplex game framework, leveraging composite regression reduction and high-accuracy convex minimization oracles. The resulting algorithm computes a (1−ε)-approximate solution in time Õ(mk/ε), where m is the number of edges and k the number of commodities—constituting the fastest known approximation for directed-graph multicommodity flow. It is also the first high-accuracy, nearly linear-time ℓ_{q,p}-flow solver, substantially improving upon prior state-of-the-art bounds.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning on the Edge & Model Compression

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We provide $m^{1+o(1)}kepsilon^{-1}$-time algorithms for computing multiplicative $(1 - epsilon)$-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 $ell_{q, p}$-regression problems to high accuracy. This method makes $ ilde{O}_{q, p}(k)$ queries to a high accuracy convex minimization oracle for an individual block, where $ ilde{O}_{q, p}(cdot)$ hides factors depending only on $q$, $p$, or $mathrm{poly}(log m)$, improving upon the $ ilde{O}_{q, p}(k^2)$ bound of [Chen-Ye, ICALP 2024]. As a result, we obtain the first almost-linear time algorithm that solves $ell_{q, p}$ flows on directed graphs to high accuracy. Second, we present optimization tools to reduce approximately solving composite $ell_{1, infty}$-regression problems to solving $m^{o(1)}epsilon^{-1}$ instances of composite $ell_{q, p}$-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.

Problem

• Efficiently solving multi-commodity flow problems on directed graphs
• Developing improved optimization methods for high-accuracy regression problems
• Reducing composite regression problems to simpler instances for faster solutions

Innovation

• Improved optimization for ell_{q, p}-regression problems
• Reduction to composite ell_{q, p}-regression instances
• Almost-linear time directed graph flow algorithm