Maximum Flow and Minimum-Cost Flow in Almost-Linear Time [article]

Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, Sushant Sachdeva
2022 arXiv   pre-print
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 that
more » ... 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.
arXiv:2203.00671v2 fatcat:hwdtmht5mjc4fn73pb3zg5tyte