Fixed-parameter tractability of directed multiway cut parameterized by the size of the cutset

Given a directed graph G, a set of k terminals and an integer p, the Directed Vertex Multiway Cut problem asks if 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 indeed are 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 (Theor. Comp. Sci. 2006) showed that in undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized by p. Marx and Razgon (STOC 2011) 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 22O(p)nO(1), i.e., FPT parameterized by size p of the cutset of the solution. This answers an open question raised by Marx (Theor. Comp. Sci. 2006) and Marx and Razgon (STOC 2011). It follows from our result that Directed Multicut is FPT for the case of k = 2 terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011).