Let G=(V,E) be a graph on n vertices and R be a set of pairs of vertices in V called \emph{requests}. A \emph{multicut} is a subset F of E such that every request xy of R is cut by F, \i.e. every xy-path of G intersects F. We show that there exists an O(f(k)nc) algorithm which decides if there exists a multicut of size at most k. In other words, the \M{} problem parameterized by the solution size k is Fixed-Parameter Tractable. The proof extends to vertex multicuts.
@article{arxiv.1010.5197,
title = {Multicut is FPT},
author = {Nicolas Bousquet and Jean Daligault and Stéphan Thomassé},
journal= {arXiv preprint arXiv:1010.5197},
year = {2010}
}