  We discuss Kondo behavior of a conduction electron system coupled with local
optical phonon by analyzing the Anderson-Holstein model with the use of a
numerical renormalization group (NRG) method. There appear three typical
regions due to the balance between Coulomb interaction $U_{\rm ee}$ and
phonon-mediated attraction $U_{\rm ph}$. For $U_{\rm ee}>U_{\rm ph}$, we
observe the standard Kondo effect concerning spin degree of freedom. Since the
Coulomb interaction is effectively reduced as $U_{\rm ee}-U_{\rm ph}$, the
Kondo temperature $T_{\rm K}$ is increased when $U_{\rm ph}$ is increased. On
the other hand, for $U_{\rm ee}<U_{\rm ph}$, there occurs the Kondo effect
concerning charge degree of freedom, since vacant and double occupied states
play roles of pseudo-spins. Note that in this case, $T_{\rm K}$ is decreased
with the increase of $U_{\rm ph}$. Namely, $T_{\rm K}$ should be maximized for
$U_{\rm ee} \approx U_{\rm ph}$. Then, we analyze in detail the Kondo behavior
at $U_{\rm ee}=U_{\rm ph}$, which is found to be explained by the polaron
Anderson model with reduced hybridization of polaron and residual repulsive
interaction among polarons. By comparing the NRG results of the polaron
Anderson model with those of the original Anderson-Holstein model, we clarify
the Kondo behavior in the competing region of $U_{\rm ee} \approx U_{\rm ph}$.
