A logical approach to multicut problems

Multicut problems are well-studied NP-complete problems in the field of network theory. Previously, by using graph theoretic methods, they have been shown to be fixed parameter tractable for different combinations of parameters, but not for any single parameter. In this paper different versions of the multicut problem are expressed in Monadic Second Order Logic (MSO) and an extended version of Courcelle's Theorem due to Arnborg, Lagergren and Seese is used to demonstrate that these problems are fixed parameter tractable with respect to the parameter ω∗, the treewidth of the input structure. Here, the input structure consists of a set V of vertices with two relations, the edge relation E of the input graph G=(V,E), and a relation H encoding all pairs of vertices to be disconnected. The contribution of this paper is two-fold: to introduce a single parameter for which the major variants of the multicut problem are fixed parameter tractable, and to use multicut problems as examples for demonstrating fruitful practical applications of logical properties and results in network theory.