  We present a general method of designing fast approximation algorithms for
cut-based minimization problems in undirected graphs. In particular, we develop
a technique that given any such problem that can be approximated quickly on
trees, allows approximating it almost as quickly on general graphs while only
losing a poly-logarithmic factor in the approximation guarantee.
  To illustrate the applicability of our paradigm, we focus our attention on
the undirected sparsest cut problem with general demands and the balanced
separator problem. By a simple use of our framework, we obtain poly-logarithmic
approximation algorithms for these problems that run in time close to linear.
  The main tool behind our result is an efficient procedure that decomposes
general graphs into simpler ones while approximately preserving the cut-flow
structure. This decomposition is inspired by the cut-based graph decomposition
of R\"acke that was developed in the context of oblivious routing schemes, as
well as, by the construction of the ultrasparsifiers due to Spielman and Teng
that was employed to preconditioning symmetric diagonally-dominant matrices.
