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  1. Free, publicly-accessible full text available January 1, 2025
  2. Free, publicly-accessible full text available November 6, 2024
  3. 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 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. 
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  6. Abstract—We develop a framework for graph sparsification and sketching, based on a new tool, short cycle decomposition – a decomposition of an unweighted graph into an edge-disjoint collection of short cycles, plus a small number of extra edges. A simple observation gives that every graph G on n vertices with m edges can be decomposed in O(mn) time into cycles of length at most 2logn, and at most 2n extra edges. We give an m1+o(1) time algorithm for constructing a short cycle decomposition, with cycles of length no(1), and n1+o(1) extra edges. Both the existential and algorithmic variants of this decomposition enable us to make progress on several open problems in randomized graph algorithms. 
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