We provide an algorithm which, with high probability, maintains a $(1-\epsilon)$-approximate maximum flow on an undirected graph undergoing $m$-edge additions in amortized $m^{o(1)} \epsilon^{-3}$ time per update. To obtain this result, we provide a more general algorithm that solves what we call the incremental, thresholded $p$-norm flow problem that asks to determine the first edge-insertion in an undirected graph that causes the minimum $\ell_p$-norm flow to decrease below a given threshold in value. Since we solve this thresholded problem, our data structure succeeds against an adaptive adversary that can only see the data structure's output. Furthermore, since our algorithm holds for $p = 2$, we obtain improved algorithms for dynamically maintaining the effective resistance between a pair of vertices in an undirected graph undergoing edge insertions. Our algorithm builds upon previous dynamic algorithms for approximately solving the minimum-ratio cycle problem that underlie previous advances on the maximum flow problem [Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva, FOCS '22] as well as recent dynamic maximum flow algorithms [v.d.Brand-Liu-Sidford, STOC '23]. Instead of using interior point methods, which were a key component of these recent advances, our algorithm uses an optimization method based on $\ell_p$-norm iterative refinement and the multiplicative weight update method. This ensures a monotonicity property in the minimum-ratio cycle subproblems that allows us to apply known data structures and bypass issues arising from adaptive queries.

翻译：我们提出一种算法，能够以高概率维护无向图在经历$m$条边添加时的$(1-\epsilon)$-近似最大流，每次更新的均摊时间为$m^{o(1)} \epsilon^{-3}$。为实现这一结果，我们提供了一个更通用的算法，解决所谓的增量阈值化$p$-范数流问题：在无向图中确定导致最小$\ell_p$-范数流首次低于给定阈值的边插入操作。由于我们解决了这个阈值化问题，数据结构能够成功应对仅能观察到其输出的自适应对手。此外，由于我们的算法适用于$p = 2$，我们获得了动态维护无向图边插入过程中一对顶点间有效电阻的改进算法。该算法建立在先前用于近似求解最小比率环问题的动态算法之上——这些算法构成了最大流问题最新进展的基础[Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva, FOCS '22]，并整合了近期的动态最大流算法[v.d.Brand-Liu-Sidford, STOC '23]。与使用作为这些近期进展关键组成部分的内点法不同，我们的算法采用基于$\ell_p$-范数迭代细化与乘法权重更新方法的优化策略。这确保最小比率环子问题具有单调性，使我们能够应用已知数据结构并绕过自适应查询引发的难题。