We provide faster strongly polynomial time algorithms solving maximum flow in structured $n$-node $m$-arc networks. Our results imply an $n^{\omega + o(1)}$-time strongly polynomial time algorithms for computing a maximum bipartite $b$-matching where $\omega$ is the matrix multiplication constant. Additionally, they imply an $m^{1 + o(1)} W$-time algorithm for solving the problem on graphs with a given tree decomposition of width $W$. We obtain these results by strengthening and efficiently implementing an approach in Orlin's (STOC 2013) state-of-the-art $O(mn)$ time maximum flow algorithm. We develop a general framework that reduces solving maximum flow with arbitrary capacities to (1) solving a sequence of maximum flow problems with polynomial bounded capacities and (2) dynamically maintaining a size-bounded supersets of the transitive closure under arc additions; we call this problem \emph{incremental transitive cover}. Our applications follow by leveraging recent weakly polynomial, almost linear time algorithms for maximum flow due to Chen, Kyng, Liu, Peng, Gutenberg, Sachdeva (FOCS 2022) and Brand, Chen, Kyng, Liu, Peng, Gutenberg, Sachdeva, Sidford (FOCS 2023), and by developing incremental transitive cover data structures.

翻译：我们为结构化$n$节点$m$弧网络中的最大流问题提供了更快的强多项式时间算法。我们的结果意味着计算最大二分图$b$-匹配的强多项式时间算法复杂度为$n^{\omega + o(1)}$，其中$\omega$是矩阵乘法常数。此外，对于具有给定树宽$W$的树分解图，该结果意味着算法时间复杂度为$m^{1 + o(1)} W$。这些成果是通过强化并高效实现Orlin（STOC 2013）提出的最先进$O(mn)$时间最大流算法中的方法而获得的。我们开发了一个通用框架，将任意容量下的最大流求解问题归约为：（1）求解一系列具有多项式有界容量的最大流子问题；（2）在弧添加操作下动态维护规模有界的传递闭包超集——我们将该问题称为增量传递闭包覆盖。我们的应用基于以下两点实现：一是利用Chen、Kyng、Liu、Peng、Gutenberg、Sachdeva（FOCS 2022）以及Brand、Chen、Kyng、Liu、Peng、Gutenberg、Sachdeva、Sidford（FOCS 2023）提出的近期弱多项式近似线性时间最大流算法；二是开发了增量传递闭包覆盖数据结构。