Multilevel Graph Partitioning Schemes

In this paper we present experiments with a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph. We investigate the effectiveness of many different choices for all three phases: coarsening, partition of the coarsest graph, and refinement. In particular, we present a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of the size of the final partition obtained after multilevel refinement. We also present a new scheme for refining during uncoarsening that is much faster than the Kernighan-Lin refinement. We test our scheme on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that our scheme consistently produces partitions that are better than those produced by spectral partitioning schemes in substantially smaller timer (10 to 35 times faster than multilevel spectral bisection). Also, when our scheme is used to compute fill reducing orderings for sparse matrices, it substantially outperforms the widely used multiple minimum degree