1 research outputs found
Flows in Almost Linear Time via Adaptive Preconditioning
We present algorithms for solving a large class of flow and regression
problems on unit weighted graphs to accuracy in
almost-linear time. These problems include -norm minimizing flow for
large (), and their duals,
-norm semi-supervised learning for close to .
As tends to infinity, -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 and norm objectives, and
decomposing graphs into uniform expanders