Fixed-parameter tractability of multicut parameterized by the size of the cutset

Given an undirected graph $G$, a collection {(s1,t1), ..., (sl,tl)} of pairs of vertices, and an integer p, the Edge Multicut problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. Vertex Multicut is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2O(p3) ⋅ nO(1), i.e., fixed-parameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f(p) ⋅ nO(1) exists for the directed version of the problem, as we show it to be W[1]-hard parameterized by the size of the cutset.