Maximum Flow by Augmenting Paths in n^2+o (1) Time

We present a combinatorial algorithm for computing exact maximum flows in directed graphs with n vertices and edge capacities from \1, , U\ in n^2+o (1) U time, which is almost optimal in dense graphs. Our algorithm is a novel implementation of the classical augmenting-path framework; we list augmenting paths more efficiently using a new variant of the push-relabel algorithm that uses additional edge weights to guide the algorithm, and we derive the edge weights by constructing a directed expander hierarchy. Even in unit-capacity graphs, this breaks the long-standing O (m\m, n^{2/3\}) time bound of the previous combinatorial algorithms by Karzanov (1973) and Even and Tarjan (1975) when the graph has m= (n^4/3) edges. Notably, our approach does not rely on continuous optimization nor heavy dynamic graph data structures, both of which are crucial in the recent developments that led to the almost-linear time algorithm by Chen et al. (FOCS 2022). Our running time also matches the n^2+o (1) time bound of the independent combinatorial algorithm by Chuzhoy and Khanna (STOC 2024) for computing the maximum bipartite matching, a special case of maximum flow.