Algorithms for Graph Partitioning on the Planted Partition Model

The NP-hard graph bisection problem is to partition the nodes of an undirected graph into two equal-sized groups so as to minimize the number of edges that cross the partition. The more general graph l-partition problem is to partition the nodes of an undirected graph into l equal-sized groups so as to minimize the total number of edges that cross between groups. We present a simple, linear-time algorithm for the graph l-partition problem and analyze it on a random \planted l-partition" model. In this model, the n nodes of a graph are partitioned into l groups, each of size n=l; two nodes in the same group are connected by an edge with some probability p, and two nodes in diierent groups are connected by an edge with some probability r < p. We show that if p ? r n ?1=2+ for some constant , then the algorithm nds the optimal partition with probability 1 ? exp(?n ()).