  In the study of correlated systems, approximations based on the dynamical
mean-field theory (DMFT) provide a practical way to take local vertex
corrections into account, which capture, respectively, particle-particle
screening at weak coupling and the formation of the local moment at strong
coupling. We show that in both limits the local vertex corrections can be
efficiently parameterized in terms of single-boson exchange, such that the
two-particle physics described by DMFT and its diagrammatic extensions is
recovered to good approximation and at a reduced computational cost. Our
investigation highlights the importance of the frequency-dependent
fermion-boson coupling (Hedin vertex) for local vertex corrections. Namely, at
weak coupling the fermion-spin-boson coupling suppresses the N\'eel temperature
of the DMFT approximation compared to the static mean-field, whereas for large
interaction it facilitates a huge enhancement of local spin-fluctuation
exchange, giving rise to the effective-exchange energy scale $4t^2/U$. We find
that parameterizations of the vertex which neglect the nontrivial part of the
fermion-boson coupling fail qualitatively at strong coupling.
