Bisections above Tight Lower Bounds

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most one, and the size of the bisection is the number of edges which go across the two parts. Every graph with m edges has a bisection of size at least [m/2], and this bound is sharp for infinitely many graphs. Therefore, Gutin and Yeo considered the parameterized complexity of deciding whether an input graph with m edges has a bisection of size at least [m/2] + k, where k is the parameter. They showed fixed-parameter tractability of this problem, and gave a kernel with O(k(2)) vertices. Here, we improve the kernel size to O(k) vertices. Under the Exponential Time Hypothesis, this result is best possible up to constant factors.