Imbalance is fixed parameter tractable

Abstract

In the Imbalance Minimization problem we are given a graph G = (V, E) and an integer b and asked whether there is an ordering v1 . . . vn of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex vi is the absolute value of the difference between the number of neighbors to the left and right of vi. The problem is also known as the Balanced Vertex Ordering problem and it finds many applications in graph drawing. We show that this problem is fixed parameter tractable and provide an algorithm that runs in time 2 O(b log b) · n O(1). This resolves an open problem of K´ara et al. [COCOON 2005].