Minimum Bisection is Fixed Parameter Tractable

In the classic MINIMUM BISECTION problem we are given as input a 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 11,41-1B11 < 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 0(2 (k3)n3 log3 n). This is the first fixed parameter tractable algorithm for MINIMUM BISECTION. 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 cut 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 for a bisection of minimum weight among solutions that contain at most k edges.