Balanced Graph Partitions

We consider the problem of partitioning a graph into $k$ components of roughly equal size while minimizing the capacity of the edges between different components of the cut. In particular we require that for a parameter $\nu \ge 1$, no component contains more than $\nu \cdot \frac{n}{k}$ of the graph vertices.

For $k=2$ and $\nu =1$ this problem is equivalent to the well known Minimum Bisection Problem for which an approximation algorithm with a polylogarithmic approximation guarantee has been presented in [FK02]. For arbitrary $k$ and $\nu \ge 2$ a bicriteria approximation ratio of $O(logn)$ was obtained by [ENRS99] using the spreading metrics technique.

We present a bicriteria approximation algorithm that for any constant $\nu >1$ runs in polynomial time and guarantees an approximation ratio of $O\left({log}^{1.5}n\right)$ (for a precise statement of the main result see \lref{Theorem}{Thm:main}). For $\nu =1$ and $k\ge 3$ we show that no polynomial time approximation algorithm can guarantee a finite approximation ratio unless $P=\mathrm{NP}$.