Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow/circulation X on a directed graph G into weighted source-to-sink paths whose weighted sum equals X. We show that, for acyclic graphs, considering the width of the graph (the minimum number of paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the version of the problem that uses only non-negative weights, we identify and characterise a new class of width-stable graphs, for which a popular heuristic is a O(log Val (X))-approximation (Val(X) being the total flow of X), and strengthen its worst-case approximation ratio from \(\Omega (\sqrt {m})\) to Ω (m/log m) for sparse graphs, where m is the number of edges in the graph. We also study a new problem on graphs with cycles, Minimum Cost Circulation Decomposition (MCCD), and show that it generalises MFD through a simple reduction. For the version allowing also negative weights, we give a (⌈ log ‖ X ‖ ⌉ +1)-approximation (‖ X ‖ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary circulations (‖ X ‖ ≤ 1), using a generalised notion of width for this problem. Finally, we disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.

最小流分解（MFD）是一个 NP 难题，即找到有向图G上的网络流/循环X的最小分解为加权源到汇路径，其加权和等于X。我们表明，对于非循环图，考虑图的宽度（覆盖其所有边所需的最小路径数）可以提高我们对其近似性的理解。对于仅使用非负权重的问题版本，我们识别并描述了一类新的宽度稳定图，其中流行的启发式是O (log瓦尔( X ))-近似 (瓦尔( X ) 是X )的总流量，并将稀疏图的最坏情况近似比从 \(\Omega (\sqrt {m})\) 增强到 Ω ( m /log m )，其中m是数字图中的边数。我们还研究了带有循环的图上的一个新问题，即最小成本循环分解（MCCD），并表明它通过简单的简化概括了 MFD。对于还允许负权重的版本，我们使用二次方方法并结合奇偶校验给出 (⌈ log ‖ X ‖ ⌉ +1) 近似（‖ X ‖ 是任何边上X的最大绝对值）使用该问题的广义宽度概念固定参数并分解酉循环 (‖ X ‖ ≤ 1)。最后，我们反驳了 Kloster 等人提出的关于最小（非负）流分解的线性独立性的猜想。[2018]，但表明其有用的含义（将权重分配给给定路径集以分解流的多项式时间分配）适用于负版本。