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Almost-Linear-Time Algorithms for Maximum Flow and Minimum-Cost Flow
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DOI：10.1145/3610940.png)
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Abstract
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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 mo(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 edgeseparable 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.
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