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 superposition equals $X$. We show that, for acyclic graphs, considering the \emph{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 \emph{width-stable} graphs, for which a popular heuristic is a \gwsimple-approximation ($|X|$ being the total flow of $X$), and strengthen its worst-case approximation ratio from $\Omega(\sqrt{m})$ to $\Omega(m / \log m)$ 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 $(\lceil \log \Vert X \Vert \rceil +1)$-approximation ($\Vert X \Vert$ 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 ($\Vert X \Vert \leq 1$), 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. [ALENEX 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.

翻译：最小流分解（MFD）是NP困难问题，旨在寻找有向图$G$上网络流/环流$X$的最小分解，将其分解为带权源-汇路径的叠加。我们证明，对于无环图，考虑图的\textit{宽度}（覆盖所有边所需的最小路径数）能推进对其近似性的理解。针对仅使用非负权重的版本，我们识别并刻画了一类新的\textit{宽度稳定}图，其中一种流行启发式算法是$|X|$的简单近似（$|X$为$X$的总流量），并将其最坏情况近似比从$\Omega(\sqrt{m})$改进至稀疏图的$\Omega(m / \log m)$，其中$m$为图的边数。我们还研究了含圈图上的新问题——最小费用环流分解（MCCD），并通过简单归约证明其推广了MFD。针对允许负权重的版本，我们采用2的幂次方法结合奇偶性修正论证与广义宽度概念，给出了$(\lceil \log \Vert X \Vert \rceil +1)$-近似（$\Vert X \Vert$为$X$在任意边上的最大绝对值）及单位环流（$\Vert X \Vert \leq 1$）的分解。最后，我们反驳了Kloster等人[ALENEX 2018]提出的关于最小（非负）流分解线性无关性的猜想，但证明其有用推论（给定路径集上多项式时间赋权以分解流）在负权重版本中成立。