We give an algorithm that, with high probability, maintains a $(1-ε)$-approximate $s$-$t$ maximum flow in undirected, uncapacitated $n$-vertex graphs undergoing $m$ edge insertions in $\tilde{O}(m+ n F^*/ε)$ total update time, where $F^{*}$ is the maximum flow on the final graph. This is the first algorithm to achieve polylogarithmic amortized update time for dense graphs ($m = Ω(n^2)$), and more generally, for graphs where $F^*= \tilde{O}(m/n)$. At the heart of our incremental algorithm is the residual graph sparsification technique of Karger and Levine [STOC '02, SICOMP '15], originally designed for computing exact maximum flows in the static setting. Our main contributions are (i) showing how to maintain such sparsifiers for approximate maximum flows in the incremental setting and (ii) generalizing the cut sparsification framework of Fung et al. [STOC '11, SICOMP '19] from undirected graphs to balanced directed graphs.

翻译：我们提出一种算法，能以高概率在 $\tilde{O}(m+ n F^*/ε)$ 总更新时间内，维护经历 $m$ 条边插入操作的无向、无容量 $n$ 顶点图中的 $(1-ε)$-近似 $s$-$t$ 最大流，其中 $F^{*}$ 是最终图上的最大流。这是首个在稠密图（$m = Ω(n^2)$）以及更一般地满足 $F^*= \tilde{O}(m/n)$ 的图上实现多对数级均摊更新时间的算法。该增量式算法的核心是 Karger 和 Levine [STOC '02, SICOMP '15] 提出的残差图稀疏化技术——该技术最初用于静态场景下精确最大流的计算。我们的主要贡献在于：(i) 展示了如何在增量式场景下维护此类稀疏化结构以实现近似最大流；(ii) 将 Fung 等人 [STOC '11, SICOMP '19] 的割稀疏化框架从无向图推广至平衡有向图。