“Maximum Flow and Minimum-Cost Flow in Almost-Linear Time”, Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, Sushant Sachdeva2022-03-01 (algorithms; similar)⁠:

[media] 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^1+o(1) time using a dynamic data structure [low-stretch spanning tree].

Our framework extends to an algorithm running in m^1+o(1) time for computing flows that minimize general edge-separable convex functions to high accuracy.

This gives an almost-linear time algorithm for several problems including entropy-regularized optimal transport, matrix scaling, p-norm flows, and isotonic regression.

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