# Flows in Almost Linear Time via Adaptive Preconditioning

## TL;DR

This paper introduces nearly linear time algorithms for solving a broad class of flow and regression problems on graphs, extending techniques from linear systems to convex objectives like _p-norm flows and total variation minimization.

## Contribution

It develops adaptive non-linear preconditioning and graph decomposition methods, enabling almost-linear time solutions for complex convex optimization problems on graphs.

## Key findings

- Achieved almost-linear time algorithms for _p-norm flow and semi-supervised learning.
- Provided new approximation methods for max-flow and total variation minimization.
- Extended linear system techniques to a wider class of convex objectives.

## Abstract

We present algorithms for solving a large class of flow and regression problems on unit weighted graphs to $(1 + 1 / poly(n))$ accuracy in almost-linear time. These problems include $\ell_p$-norm minimizing flow for $p$ large ($p \in [\omega(1), o(\log^{2/3} n) ]$), and their duals, $\ell_p$-norm semi-supervised learning for $p$ close to $1$. As $p$ tends to infinity, $\ell_p$-norm flow and its dual tend to max-flow and min-cut respectively. Using this connection and our algorithms, we give an alternate approach for approximating undirected max-flow, and the first almost-linear time approximations of discretizations of total variation minimization objectives. This algorithm demonstrates that many tools previous viewed as limited to linear systems are in fact applicable to a much wider range of convex objectives. It is based on the the routing-based solver for Laplacian linear systems by Spielman and Teng (STOC '04, SIMAX '14), but require several new tools: adaptive non-linear preconditioning, tree-routing based ultra-sparsification for mixed $\ell_2$ and $\ell_p$ norm objectives, and decomposing graphs into uniform expanders.