Multicut is FPT

TL;DR

This paper proves that the Multicut problem, which involves removing edges or vertices to separate specified pairs of vertices, is fixed-parameter tractable when parameterized by the size of the solution, using an algorithm with runtime O(f(k)n^c).

Contribution

The authors establish that Multicut is fixed-parameter tractable, providing an algorithm with runtime depending on the parameter k and polynomial in the size of the graph, extending to vertex multicuts.

Findings

Existence of an FPT algorithm for Multicut parameterized by solution size k.

Algorithm runs in O(f(k)n^c) time, confirming fixed-parameter tractability.

Extension of results to vertex multicuts.

Abstract

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 $O(f(k)n^c)$ 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.