We introduce the concept of low-step multi-commodity flow emulators for any undirected, capacitated graph. At a high level, these emulators contain approximate multi-commodity flows whose paths contain a small number of edges, shattering the infamous flow decomposition barrier for multi-commodity flow. We prove the existence of low-step multi-commodity flow emulators and develop efficient algorithms to compute them. We then apply them to solve constant-approximate $k$-commodity flow in $O((m+k)^{1+\epsilon})$ time. To bypass the $O(mk)$ flow decomposition barrier, we represent our output multi-commodity flow implicitly; prior to our work, even the existence of implicit constant-approximate multi-commodity flows of size $o(mk)$ was unknown. Our results generalize to the minimum cost setting, where each edge has an associated cost and the multi-commodity flow must satisfy a cost budget. Our algorithms are also parallel.

翻译：本文针对任意无向带容量图，提出了低步数多商品流模拟器的概念。从高层次看，这类模拟器包含路径仅由少量边构成的近似多商品流，从而突破了多商品流问题中著名的流分解障碍。我们证明了低步数多商品流模拟器的存在性，并开发了高效算法对其进行计算。随后将其应用于求解常数近似的$k$商品流问题，时间复杂度为$O((m+k)^{1+\epsilon})$。为规避$O(mk)$的流分解障碍，我们采用隐式表示法输出多商品流；在本研究之前，即使规模为$o(mk)$的隐式常数近似多商品流的存在性亦属未知。我们的结果可推广至最小代价场景，其中每条边具有关联代价且多商品流需满足代价约束。所提算法同样具备并行计算能力。