Maximum Flow and Minimum-Cost Flow in Almost-Linear Time

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 cycle...

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Bibliographic Details
Published inProceedings / annual Symposium on Foundations of Computer Science pp. 612 - 623
Main Authors Chen, Li, Kyng, Rasmus, Liu, Yang P., Peng, Richard, Gutenberg, Maximilian Probst, Sachdeva, Sushant
Format Conference Proceeding
LanguageEnglish
Published IEEE 01.10.2022
Subjects
Approximation algorithms
Computer science
Convex optimization
Costs
Data structures
Directed acyclic graph
Directed graphs
Heuristic algorithms
Integral equations
Interior point methods
Maximum flow
Minimum cost flow
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ISSN2575-8454
DOI10.1109/FOCS54457.2022.00064

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Summary: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.
ISSN:2575-8454
DOI:10.1109/FOCS54457.2022.00064