Balanced graph partitioning

In this paper we consider the problem of (k, ν)-balanced graph partitioning - dividing the vertices of a graph into k almost equal size components (each of size less than ν · n/k) so that the capacity of edges between different components is minimized. This problem is a natural generalization of several other problems such as minimum bisection, which is the (2, 1)-balanced partitioning problem. We present a bicriteria polynomial time approximation algorithm with an O(log² n)-approximation for any constant ν > 1. For ν = 1 we show that no polytime approximation algorithm can guarantee a finite approximation ratio unless P = NP. Previous work has only considered the (k, ν)-balanced partitioning problem for ν ≥ 2.