  Using ultrafast optical techniques, we probe the hole-doping dependence of
the electron-boson coupling constant $\lambda$ in
Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$. In the overdoped region, we observe a
correlation between ($\lambda$) and the superconducting transition temperature
$T_{c}$. Upon performing the McMillan analysis, however, we find that $\lambda$
is too small to explain the high $T_{c}$'s, and that the Coulomb
pseudopotential $\mu^{\ast}$ is negative. Our analysis therefore reveals two
components in the mechanism of high-$T_{c}$ superconductivity -- a dominant
pre-existing pairing interaction, together with a weaker electron-phonon
interaction that fine-tunes $T_{c}$.
