greedy algorithm for Maximum directed cut
The starting point is the trivial random algorithm that chooses $S$ completely at random. Each directed edge is cut with probability $1/4$ (why?), and so in expectation, this random algorithm gives a $1/4$ approximation.

We can derandomize this algorithm using the method of conditional expectations. Arrange the points in order: $1,\ldots,n$. At step $i$, we know which of $1,\ldots,i-1$ are in $S$ and which are in $\bar{S}$. If we put $i \in S$, then we can compute the expected cost of the output given that all further choices are made randomly; and we can do the same if we put $i \in \bar{S}$. One of these choices cuts at least $1/4$ of the edges (in terms of weight), and this is the one we choose.

We can further optimize this algorithm. Consider the following two expressions:
$$
A = \sum_{\substack{j&lt;i\\j\in\bar{S}}} w_{ij} + \frac{1}{2} \sum_{j&gt;i} w_{ij}, \\
B = \sum_{\substack{j&lt;i\\j\in S}} w_{ji} + \frac{1}{2} \sum_{j&gt;i} w_{ji}.
$$
(Here $w_{ij}$ is the cost of the edge from $i$ to $j$.)
If $A&gt;B$ then the derandomized algorithm puts $i$ in $S$, whereas if $A&lt;B$ it puts $i$ in $\bar{S}$ (if $A=B$ it doesn't matter). We can perhaps think of this algorithm as a greedy algorithm.



As an aside, the so-called double greedy algorithm gives a $1/2$ approximation algorithm. The greedy algorithm you suggest in the comments is good in combination with the random algorithm: if you run both and choose the better solution, you get a $2/5$ approximation (see Feige's paper). The semidefinite relaxation gives the (probably) optimal approximation ratio, which is roughly 0.874.