Maximum Flow and Minimum-Cost Flow in Almost-Linear Time

We present 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.