Balanced Graph Partitions

In this paper we consider the problem of $(k,\nu )$-balanced graph partitioning - dividing the vertices of a graph into $k$ almost equal size components (each of size less than $\nu \cdot \frac{n}{k}$) 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 $O\left({log}^{2}n\right)$-approximation for any constant $\nu >1$. For $\nu =1$ we show that no polytime approximation algorithm can guarantee a finite approximation ratio unless $P=\mathrm{NP}$. Previous work has only considered the $(k,\nu )$-balanced partitioning problem for $\nu \ge 2$.