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).

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 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 Multicut problem parameterized by the solution size k is Fixed-Parameter Tractable.

The {\sc Multicut In Trees} problem consists in deciding, given a tree, a set of requests (i.e. paths in the tree) and an integer $k$, whether there exists a set of $k$ edges cutting all the requests. This problem was shown to be FPT by Guo and Niedermeyer (2005). They also provided an exponential kernel. They asked whether this problem has a polynomial kernel. This question was also raised by Fellows (2006). We show that {\sc Multicut In Trees} has a polynomial kernel.