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 $\ell_{q, p}$-norm multicommodity flow problem to a $(1 + \varepsilon)$ approximation in time $O_{q, p}(m^{1+o(1)} k^2 \log(1 / \varepsilon))$, where $k$ is the number of commodities, and $O_{q, p}(\cdot)$ hides constants depending only on $q$ or $p$. As $q$ and $p$ approach to $1$ and infinity respectively, $\ell_{q, p}$-norm flow tends to maximum concurrent flow. We introduce the first iterative refinement framework for $\ell_{q, p}$-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.

翻译：多商品流问题是网络流和组合优化中的经典问题，在交通运输、通信、物流和供应链管理等领域具有广泛应用。现有算法通常聚焦于低精度近似解，而高精度算法则普遍依赖通用线性规划求解器。本文针对无向图上一大类多商品流问题提出了高效的高精度算法，相比通用线性规划求解器实现了更优的运行时间。我们的主要结果表明：对于$\ell_{q, p}$-范数多商品流问题，可在$O_{q, p}(m^{1+o(1)} k^2 \log(1 / \varepsilon))$时间内达到$(1 + \varepsilon)$近似精度，其中$k$为商品数量，$O_{q, p}(\cdot)$隐藏仅依赖于$q$或$p$的常数。当$q$和$p$分别趋近于1和无穷大时，$\ell_{q, p}$-范数流趋于最大并发流。我们首次提出针对$\ell_{q, p}$-范数最小化问题的迭代精化框架，将原问题简化为求解一系列可分解的残差问题。在$k$-商品流情形下，每个残差问题可分解为$k$个单商品凸流问题，每个子问题均可在近线性时间内求解。由于多商品流的多个经典变体已被证明在高精度求解域中可完备表示线性规划问题[Ding-Kyng-Zhang, ICALP'22]，本研究为探索更高效的高精度多商品流算法提供了新方向。