Almost-Linear-Time Algorithms for Maximum Flow and Minimum-Cost Flow

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 m1 o(1) time. Our algorithm builds the flow through a sequence of m1 o(1) approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized mo(1) time using a new dynamic graph data structure. Our framework extends to algorithms running in m1 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. The maximum flow problem and its generalization, the minimum-cost flow problem, are classic combinatorial graph problems that find numerous applications in engineering and scientific computing.