We present the first polynomial-time algorithm for computing a near-optimal \emph{flow}-expander decomposition. Given a graph $G$ and a parameter $φ$, our algorithm removes at most a $φ\log^{1+o(1)}n$ fraction of edges so that every remaining connected component is a $φ$-\emph{flow}-expander (a stronger guarantee than being a $φ$-\emph{cut}-expander). This achieves overhead $\log^{1+o(1)}n$, nearly matching the $Ω(\log n)$ graph-theoretic lower bound that already holds for cut-expander decompositions, up to a $\log^{o(1)}n$ factor. Prior polynomial-time algorithms required removing $O(φ\log^{1.5}n)$ and $O(φ\log^{2}n)$ fractions of edges to guarantee $φ$-cut-expander and $φ$-flow-expander components, respectively.

翻译：我们提出了首个计算近似最优\emph{流}-扩展图分解的多项式时间算法。给定图$G$和参数$φ$，我们的算法最多移除$φ\log^{1+o(1)}n$比例的边，使得每个剩余连通分量均为$φ$-\emph{流}-扩展图（此保证强于$φ$-\emph{割}-扩展图条件）。该算法实现了$\log^{1+o(1)}n$的开销，几乎达到了割扩展图分解中已存在的$Ω(\log n)$图论下界，仅相差$\log^{o(1)}n$因子。此前已知的多项式时间算法分别需要移除$O(φ\log^{1.5}n)$和$O(φ\log^{2}n)$比例的边才能保证获得$φ$-割扩展图分量与$φ$-流扩展图分量。