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^O(k3)n³ log³ 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. [ABSTRACT FROM AUTHOR]