Accelerated Approximate Optimization of Multi-Commodity Flows on Directed Graphs
Li Chen, Andrei Graur, Aaron Sidford
TL;DR
This work develops near-linear time algorithms for directed multi-commodity flow (MCF) problems by introducing a unified composite optimization framework. It reduces directed MCF to composite $\ell_{1,\infty}$ and $\ell_{q,p}$ regression problems and solves them via extragradient methods with area-convex and doubly entropic regularizers, leveraging separable convex minimization oracles. The result is $(1\pm\epsilon)$-approximate solutions for concurrent, maximum, and maximum-weight MCF in time $\widehat{O}(m k \epsilon^{-1})$, enabled by solving $\widehat{O}(k)$ single-commodity convex flow subproblems per iteration and an almost-linear-time convex flow solver. A novel high-accuracy composite $\ell_{q,p}$-regression method and a refined analysis of constrained box-simplex games underlie the improvements, addressing obstacles unique to directed graphs. The framework offers a path toward optimal query and runtime complexity for a broad class of MCF instances and related regression problems, with several open questions on tighter dependencies and potential further reductions to single-commodity solves.
Abstract
We provide $m^{1+o(1)}kε^{-1}$-time algorithms for computing multiplicative $(1 - ε)$-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 $\tilde{O}_{q, p}(k)$ queries to a high accuracy convex minimization oracle for an individual block, where $\tilde{O}_{q, p}(\cdot)$ hides factors depending only on $q$, $p$, or $\mathrm{poly}(\log m)$, improving upon the $\tilde{O}_{q, p}(k^2)$ 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 $m^{o(1)}ε^{-1}$ 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.