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We give the first O(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.

Approximate Undirected Maximum Flows in O(mpolylog(n)) Time