Minimum Bisection Is Fixed-Parameter Tractable

In the classic Minimum Bisection problem we are given as input an undirected graph $G$ and an integer $k$. The task is to determine whether there is a partition of $V(G)$ into two parts $A$ and $B$ such that $||A|-|B|| \leq 1$ and there are at most $k$ edges with one endpoint in $A$ and the other in $B$. In this paper we give an algorithm for Minimum Bisection with running time $2^{\mathcal{O}(k^3)}n^3 \log^3 n$. This is the first fixed parameter tractable algorithm for Minimum Bisection parameterized by $k$. At the core of our algorithm lies a new decomposition theorem that states that every graph $G$ can be decomposed by small separators into parts where each part is “highly connected” in the following sense: any separator of bounded size can separate only a limited number of vertices from each part of the decomposition. Our techniques generalize to the weighted setting, where we seek a bisection of minimum weight among solutions that contain at most $k$ edges.