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

Tuesday, November 29, 2022 - 4:00pm to 5:15pm
Milk & Cookies served 4--4:15 pm
32-G449 (Patil/Kiva)
Yang P Liu (Stanford)

Abstract. 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 dynamic data structure. 

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. 
Joint work with Li Chen, Rasmus Kyng, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva.