Fast Spectral Graph Partitioning with a Randomized Eigensolver

A known problem in parallel computing is how to partition a matrix such that work can be distributed among several processors efficiently. One technique to do this is spectral graph partitioning, which uses the eigenvectors of the graph Laplacian to determine the optimal way for the matrix to be divided. This partitioning method is particularly suited for parallelization, specifically for GPUs, as it mainly relies on linear algebra operations. However, this increased parallelism may come at the cost of accuracy. In this work, we present a novel improvement to spectral graph partitioning by replacing the exact eigensolver (LOBPCG) with a randomized eigensolver roughly an order of magnitude faster. While the accuracy of the eigensolver is typically worse, we show that for graph partitioning this is sufficient. Our algorithm is implemented in the Sphynx spectral graph partitioner, contained in the Zoltan2 package of Trilinos. Results show this randomized method in general gives a substantial speedup with minimal loss in the quality of the edge cut. In some cases the randomized method even gives slightly better edge cuts than the LOBPCG eigensolver.