Fixed-Parameter Tractability of Multicut Parameterized by the Size of the Cutset

Given an undirected graph G, a collection {(s(1), t(1)), ... , (s(k), t(k))} 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 t(i). 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 2(O)(p(3)) . n(O(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) . n(O(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.