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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 thatarXiv:2203.00671v2 fatcat:hwdtmht5mjc4fn73pb3zg5tyte