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 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(2O(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.