We augment the time-linear formulation of the Kadanoff-Baym equations for systems of interacting electrons and quantized phonons or photons with the $G\tilde{W}$ approximation, the Coulomb interaction $\tilde{W}$ being dynamically screened by both electron-hole pairs and bosonic particles. We also show how to combine different approximations to include simultaneously multiple correlation effects in the dynamics. The final outcome is a versatile framework comprising $2^{12}$ distinct diagrammatic methods, each scaling linearly in time and preserving all fundamental conservation laws. The dramatic improvement over current state-of-the-art approximations brought about by $G\tilde{W}$ is demonstrated in a study of the correlation-induced charge migration of the glycine molecule in an optical cavity.

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• Electronic excitation & ionization
• Electronic structure
• Quasiparticles & collective excitations
• Nonequilibrium systems
• First-principles calculations
• Nonequilibrium Green's function