Approximate undirected maximum flows in 0(rapolylog(n)) time

Abstract

We give the first 0(mpolylog(n)) time algorithms for approximating maximum flows in undirected graphs and constructing polylog(n)-quality cut-approximating hierarchical tree decompositions. Our algorithm invokes existing algorithms for these two problems recursively while gradually incorporating size reductions. These size reductions are in turn obtained via ultra-sparsifiers, which are key tools in solvers for symmetric diagonally dominant (SDD) linear systems.