Complexity of the multicut problem, in its vanilla, partial and generalized versions, in graphs of bounded treewidth.

In the Multicut problem, the input consists of a graph and a set of pairs of terminal vertices that have to be disconnected by the removal of a set of edges with minimum total weight. The Partial Multicut problem generalizes Multicut by only requiring a given number of pairs of terminals to be disconnected by the removal of edges. It has been shown that Multicut remains APX-hard if the treewidth tw(G) of its input graph G is bounded, but that it is FPT w.r.t. tw(G+H), where H is the demand graph, whose vertices are the terminals and whose edges are the pairs of terminals. We prove that this also holds for Partial Multicut. Furthermore, it has been proved that Multicut also becomes polynomial-time solvable if tw(G) is bounded and H is complete (this is the Multiterminal Cut problem). We extend this result in two directions, by proving that Multicut remains polynomial-time solvable if tw(G+H¯) is bounded, and that this remains true for Partial Multicut. Finally, we show that if we further generalize the problem to allow non-unitary profits for pairs of terminals, then the problem is weakly NP-hard and has an FPTAS if tw(G+H) is bounded, and becomes APX-hard if tw(G+H¯) is bounded.