Computer Science/Discrete Mathematics Seminar II

We give the first almost-linear time algorithm for computing exact maximum flows and minimum-cost flows on directed graphs. By well known reductions, this implies almost-linear time algorithms for several problems including bipartite matching, optimal transport, and undirected vertex connectivity.

Our algorithm is designed using a new Interior Point Method (IPM) that builds the flow as a sequence of almost-linear number of approximate undirected minimum-ratio cycles, each of which is computed and processed very efficiently using a new dynamic data structure.

Our framework extends to give an almost-linear time algorithm for computing flows that minimize general edge-separable convex functions to high accuracy. This gives the first 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, Yang Liu, Richard Peng, and Maximilian Probst Gutenberg.