One-round feistel network (DES) attack
Since

$$X' = F(X, k) = P(S(E(X) \oplus k ))$$

the value of key $K$ is an element of the set

$$ \{V \oplus E(R_0): V \in S^{-1}(P^{-1}(Z)) \}$$

where 

$$Z = R_1 \oplus L_0$$

and

$$
S^{-1}(P^{-1}(Z)) 
$$
is the set of possible inverse images of the parallel map of the Sboxes $S$ which is not one to one.

Total number of possible keys is $2^{16}$ because Sboxes are not bijective, there are $4$ possible inputs leading to same output.