Before revision:  We provide an O(m)-time randomized algorithm for the \ell_2-norm flow diffusion problem, obtaining the approximation factor of 1+1/\mathrm{poly local cut structure, we give an alternative approach for finding locally-biased low conductance cuts . It is done simply by sweeping over the dual solution vector. This algorithm demonstrates a novel way of dealing with inequality constraints in graph optimization problems.
After revision:  We provide an O(m)-time randomized algorithm for the \ell_2-norm flow diffusion problem, obtaining the approximation factor of 1+1/\mathrm{poly local cut structure, we give an alternative approach for finding such cuts in nearly linear time. From a technical point of view, our algorithm contributes a novel way of dealing with inequality constraints in graph optimization problems.