The complexity of multicut and mixed multicut problems in (di)graphs
The multicut problem with respect to the fixed pattern (di)graph Q is the following: given a (di)graph G, an integer k and a 1-1 mapping ? of the vertices of Q to a subset X  of V(G) V ( G ) ; determine whether there exists a set S of at most k edges (arcs) of G  such that G?S G ? S has no (?(q),?(q?)) ( ? ( q ) , ? ( q ? ) ) -path for every edge (arc) qq? q q ? of Q. We first prove a dichotomy result for the complexity (they are all either polynomial or NP-complete) for multicut problems in (di)graphs in terms of the pattern Q. Then we consider a variant where the pattern is a mixed graph M; the input is a digraph D, an integer k and a 1-1 mapping ?  from V(M) V ( M ) to some subset of V(D) V ( D ) and the goal is to decide whether there exists a set S of at most k arcs of D  that meets every (?(q),?(q?)) ( ? ( q ) , ? ( q ? ) ) -path in D  for every arc qq? q q ? of M  and every (?(p),?(p?)) ( ? ( p ) , ? ( p ? ) ) -path in the underlying undirected graph of D  for every edge pp? p p ? of M. Again we prove a dichotomy result for the complexity of such mixed multicut problems in terms of the pattern M. It turns out that the only polynomial cases are those where the pattern is either a polynomial graph pattern or a polynomial digraph pattern. As soon as M contains both edges and arcs, the M-mixed cut problem becomes NP-complete. Finally, we prove that the directed feedback arc set problem is NP-complete, even in the class of digraphs which have a feedback vertex set of size 2.
multicut dichotomy mixed graphs feedback arc set mixed problem np-complete