Multicut Is FPT

Let $G=(V,E)$ be a graph on $n$ vertices and $R$ be a set of pairs of vertices in $V$ called requests. A multicut is a subset $F$ of $E$ such that every request $xy$ of $R$ is separated by $F$, i.e., every $xy$-path of $G$ intersects $F$. We show that there exists an $O(f(k)n^c)$ algorithm which decides if there exists a multicut of size at most $k$. In other words, the Multicut problem parameterized by the solution size $k$ is fixed-parameter tractable (FPT).