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

by Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng,
Maximilian Probst Gutenberg, and Sushant Sachdeva

#### Oded's comments

I guess (since I don't know) that this paper was featured
on all BLOGs a couple of weeks ago, and it definitely does not
need my recommendation; still, it feels odd to ignore it.
Another recommendation, which probably bursts to an open door,
is watching a talk on this work;
Part 1 (approximately 20 minutes) provides a good overview
of the context and contents of this work.

#### The original 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.

Available from
ArXiv 2203.00671.

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