Approximation algorithms for Directed Minimum Cut with Cardinality Constraints We would like to know whether there are any known approximation results for the cardinality constrained minimum $s$-$t$-cut on directed graphs. We weren't able to find any such result in literature.

The problem is defined as follows:

Instance: A directed graph $G=(V,E)$, a cost function $w : E \to \mathbb{R_0^+}$, two vertices $s,t \in V$ and an integer $k$.

Solution: An $s$-$t$-cut, i.e. a partition of $V$ into two sets $V_1, V_2$ such that $s \in V_1$, $t \in V_2$ and the number of edges that cross the cut is at most $k$, i.e. $|\{ (u,v) \in E: u \in V_1, v \in V_2 \}| \le k$.

Measure (to minimize): The cost of the cut: $$ \sum_{ (u,v) \in E : u \in V_1, v \in V_2 } w(u,v) $$

In "Cardinality constrained and multicriteria (multi)cut problems" the autors prove that this problem is strongly NP-Hard even for undirected graphs.

We are mainly interested in approximation algorithms for directed graphs, but approximation results for the undirected case might also be useful.

Thank you for any insights.
We can get a $(2,2)$ bi-criteria approximation as follows (or more generally $(1+\varepsilon, 1 + 1/\varepsilon)$ bi-criteria approximation).

We may assume that we know the cost of the optimal solution. Denote it by $OPT$. Let 
$$w'(u,v) = \frac{w(u,v)}{OPT} + \frac{1}{k}.$$
Consider the optimal solution $(V_1, V_2)$. Then
$$\sum_{(u,v) \in E(V_1, V_2)} w'(u,v) = \sum_{(u,v) \in E(V_1, V_2)} \left(\frac{w(u,v)}{OPT} + \frac{1}{k}\right) = 1 + \frac{|E(V_1,V_2)|}{k} \leq 2.$$

Our algorithm finds the minimum directed $s$-$t$ cut $(V_1', V_2')$ in $G$ with edge weights $w'$. The cost of this cut is at most $2$. Therefore, the cut $(V_1', V_2')$ cuts at most
$$E(V_1', V_2') = \sum_{(u,v)\in E(V_1',V_2') } 1 \leq k \sum_{(u,v)\in E(V_1',V_2')} w'(u,v) \leq 2k$$
edges. The cost of the cut is at most
$$\sum_{(u,v)\in E(V_1',V_2')} w(u,v) \leq OPT \sum_{(u,v)\in E(V_1',V_2')} w'(u,v) \leq 2OPT.$$