  We study the (nearly) optimal mechanisms in $(\epsilon,\delta)$-approximate
differential privacy for integer-valued query functions and vector-valued
(histogram-like) query functions under a utility-maximization/cost-minimization
framework. We characterize the tradeoff between $\epsilon$ and $\delta$ in
utility and privacy analysis for histogram-like query functions ($\ell^1$
sensitivity), and show that the $(\epsilon,\delta)$-differential privacy is a
framework not much more general than the $(\epsilon,0)$-differential privacy
and $(0,\delta)$-differential privacy in the context of $\ell^1$ and $\ell^2$
cost functions, i.e., minimum expected noise magnitude and noise power. In the
same context of $\ell^1$ and $\ell^2$ cost functions, we show the
near-optimality of uniform noise mechanism and discrete Laplacian mechanism in
the high privacy regime (as $(\epsilon,\delta) \to (0,0)$). We conclude that in
$(\epsilon,\delta)$-differential privacy, the optimal noise magnitude and noise
power are $\Theta(\min(\frac{1}{\epsilon},\frac{1}{\delta}))$ and
$\Theta(\min(\frac{1}{\epsilon^2},\frac{1}{\delta^2}))$, respectively, in the
high privacy regime.
