From Incremental Transitive Cover to Strongly Polynomial Maximum Flow

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.