宽度对流量分解的促进与阻碍作用
Width Helps and Hinders Splitting Flows
AI总结：
本文研究最小流量分解（MFD）的可近似性，通过引入图的宽度概念，改进了非负权重版本的近似比，提出新问题MCCD并给出负权重版本的近似算法，同时反驳了相关猜想。
摘要：
最小流量分解（MFD）是NP难问题，即在有向图$G$上，将网络流/环流$X$分解为加权源到汇路径的最小分解，使得这些路径的叠加等于$X$。我们证明，对于无环图，考虑图的“宽度”（覆盖其所有边所需的最少路径数）能推进对其可近似性的理解。对于仅使用非负权重的版本，我们识别并刻画了一类新的“宽度稳定”图，对于这类图，一种流行的启发式算法是$|X|$-简单近似（$|X|$为$X$的总流量），并将其最坏情况近似比从$\Omega(\sqrt{m})$加强到$\Omega(m / \log m)$（针对稀疏图，其中$m$为图的边数）。我们还研究了有环图上的一个新问题——最小费用环流分解（MCCD），并证明它通过简单归约推广了MFD。对于允许负权重的版本，我们给出一个$(\lceil \log \Vert X \Vert \rceil +1)$-近似（$\Vert X \Vert$为$X$在任何边上的最大绝对值），采用二的幂次方法，结合奇偶性修正参数和单位环流（$\Vert X \Vert \leq 1$）的分解，使用该问题的广义宽度概念。最后，我们反驳了Kloster等人[ALENEX 2018]提出的关于最小（非负）流量分解线性无关性的猜想，但证明其有用推论（多项式时间内为给定路径集分配权重以分解流量）在负权重版本中成立。
英文摘要：
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 $Ω(\sqrt{m})$ 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 $(\lceil \log \Vert X \Vert \rceil +1)$-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
