We provide $m^{1+o(1)}k\epsilon^{-1}$-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 $\tilde{O}_{q, p}(k)$ 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 $m^{o(1)}\epsilon^{-1}$ 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.

翻译：我们提出了在$m$条边的有向图上计算$k$商品多商品流问题（包括并发多商品流和最大多商品流）的乘法$(1 - \epsilon)$近似解的$m^{1+o(1)}k\epsilon^{-1}$时间算法。为获得这些结果，我们提供了具有潜在独立价值的新优化工具。首先，我们提出了一种改进的优化方法，用于高精度求解$\ell_{q, p}$回归问题。该方法对单个模块的高精度凸最小化预言机进行$\tilde{O}_{q, p}(k)$次查询（其中$\tilde{O}_{q, p}(\cdot)$隐藏了仅依赖于$q$、$p$或$\mathrm{poly}(\log m)$的因子），改进了[Chen-Ye, ICALP 2024]中$\tilde{O}_{q, p}(k^2)$的界。由此，我们首次获得了能在有向图上高精度求解$\ell_{q, p}$流的近线性时间算法。其次，我们提出了将复合$\ell_{1, \infty}$回归问题的近似求解约化为求解$m^{o(1)}\epsilon^{-1}$个复合$\ell_{q, p}$回归问题实例的优化工具。该方法基于近期在求解盒-单纯形博弈[Jambulapati-Tian, NeurIPS 2023]方面的进展，以及[Sherman, STOC 2017]中引入的区域凸正则化器，从而为该问题的约束版本获得了更快的收敛速率。精心结合这些技术最终得到了我们的有向多商品流算法。