  We present a time-linear scaling method to simulate open and correlated
quantum systems out of equilibrium. The method inherits from many-body
perturbation theory the possibility to choose selectively the most relevant
scattering processes in the dynamics, thereby paving the way to the real-time
characterization of correlated ultrafast phenomena in quantum transport. The
open system dynamics is described in terms of an embedding correlator from
which the time-dependent current can be calculated using the Meir-Wingreen
formula. We show how to efficiently implement our approach through a simple
grafting into recently proposed time-linear Green's function methods for closed
systems. Electron-electron and electron-phonon interactions can be treated on
equal footing while preserving all fundametal conservation laws.
