Algorithms for Graph Partitioning on the Planted Partition Model

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Abstract

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 different groups are connected by an edge with some probability r < p. We show that if pr > n  − − 1/2 + ε for some constant ε, then the algorithm finds the optimal partition with probability \(1 - \exp( -n^{\theta(\epsilon)})\) .