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.

翻译：我们从参数化的复杂角度研究问题 Symrimed 指向多截面 。 在这个问题中, 输入是 $D, 一组削减要求$C $( s_ 1, t_ 1),\ ldot, (s@ ell, t ⁇ ell) $ 美元和整数美元, 任务在于找到一套 $X subseete V(D) 美元, 其大小最多为 $k 美元 。 这个问题在最接近的算法研究中是已知的, 每1美元( leq i) i\leq\ leq 美元, $X$美元 的交点, 美元 或所有 $( t_i, t_ 1, t_ 1) 美元,\ lddaldcrop 的 。 我们注意到, 美元 直截面的Fxxxxxxxxxxxxx, 通过直截面的 Fxxxxxxx, 通过直截面的Fxxxxxxxxxx, 问题 。