Maximum Flow and Minimum-Cost Flow in Almost-Linear Time - Chen, Li; Kyng, Rasmus; Liu, Yang P.; Peng, Richard; Gutenberg, Maximilian Probst; Sachdeva, Sushant (Proceedings / annual Symposium on Foundations of Computer Science, 2022)
We give an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with m edges and polynomially bounded integral demands, costs, and capacities in m^{1+o(1)} time. Our algorithm builds the flow through a sequence of m^{1+o(1)} approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized m^{o(1)} time using a new dynamic graph data structure. Our framework extends to algorithms running in m^{1+o(1)} time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, p-norm flows, and p-norm isotonic regression on arbitrary directed acyclic graphs.