  We study the problem Symmetric Directed Multicut from a parameterized
complexity perspective. In this problem, the input is a digraph $D$, a set of
cut requests $C=\{(s_1,t_1),\ldots,(s_\ell,t_\ell)\}$ and an integer $k$, and
the task is to find a set $X \subseteq V(D)$ of size at most $k$ such that for
every $1 \leq i \leq \ell$, $X$ intersects either all $(s_i,t_i)$-paths or all
$(t_i,s_i)$-paths. Equivalently, every strongly connected component of $D-X$
contains at most one vertex out of $s_i$ and $t_i$ for every $i$. This problem
is previously known from research in approximation algorithms, where it is
known to have an $O(\log k \log \log k)$-approximation. We note that the
problem, parameterized by $k$, directly generalizes multiple interesting FPT
problems such as (Undirected) Vertex Multicut and Directed Subset Feedback
Vertex Set. We are not able to settle the existence of an FPT algorithm
parameterized purely by $k$, but we give three partial results: An FPT
algorithm parameterized by $k+\ell$; an FPT-time 2-approximation parameterized
by $k$; and an FPT algorithm parameterized by $k$ for the special case that the
cut requests form a clique, Symmetric Directed Multiway Cut. The existence of
an FPT algorithm parameterized purely by $k$ remains an intriguing open
possibility.
