  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.
