Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
Rajesh Chitnis, MohammadTaghi Hajiaghayi, Dániel Marx · SIAM Journal on Computing · 2013
Given a directed graph $G$, a set of $k$ terminals, and an integer $p$, the Directed Vertex Multiway Cut problem asks whether there is a set $S$ of at most $p$ (nonterminal) vertices whose removal disconnects each terminal from all other terminals. Directed Edge Multiway Cut is the analogous problem where $S$ is a set of at most $p$ edges. These two problems are indeed known to be equivalent. A natural generalization of the multiway cut is the Multicut problem, in which we want to disconnect only a set of $k$ given pairs instead of all pairs. Marx [Theoret. Comput. Sci., 351 (2006), pp. 394--406] showed that in undirected graphs Vertex/Edge Multiway cut is fixed-parameter tractable (FPT) parameterized by $p$. Marx and Razgon [Proceedings of the 43rd ACM Symposium on Theory of Computing, 2011, pp. 469--478] showed that undirected Multicut is FPT and Directed Multicut is W[1]-hard parameterized by $p$. We complete the picture here by our main result, which is that both Directed Vertex Multiway Cut and Directed Edge Multiway Cut can be solved in time $2^{2^{O(p)}}n^{O(1)}$, i.e., FPT parameterized by size $p$ of the cutset of the solution. This answers an open question raised by the aforementioned papers. It follows from our result that Directed Edge/Vertex Multicut is FPT for the case of $k=2$ terminal pairs, which answers another open problem raised by Marx and Razgon.