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We give the first O(m polylog(n)) time algorithms for approximating maximumflows in undirected graphs and constructing polylog(n) -qualitycut-approximating hierarchical tree decompositions. Our algorithm invokesexisting algorithms for these two problems recursively while graduallyincorporating size reductions. These size reductions are in turn obtained viaultra-sparsifiers, which are key tools in solvers for symmetric diagonallydominant (SDD) linear systems.

Approximate Undirected Maximum Flows in O(m polylog(n)) Time