1/2 Approximation to MAX-DICUT by rounding a linear program
The constraints force $z_{ij} = \min(x_i,1-x_j)$. Therefore you have to show that
$$
\left(\frac{1}{4} + \frac{x_i}{2}\right) \left(\frac{1}{4} + \frac{1-x_j}{2}\right) \geq \frac{1}{2} \min(x_i,1-x_j).
$$
By symmetry, we can assume that $x_i \leq 1 - x_j$, and so we have to show that
$$
\left(\frac{1}{4} + \frac{x_i}{2}\right)^2 \geq \frac{x_i}{2}.
$$
You can check that this holds for all $x_i$. Equality holds when $x_i = 1-x_j = 1/2$, showing that the approximation ratio $1/2$ is tight (for the analysis).