Exact k-way sparse matrix partitioning

Abstract

To minimize the communication in parallel sparse matrix-vector multiplication while maintaining load balance, we need to partition the sparse matrix optimally into k disjoint parts, which is an NP-complete problem. We present an exact algorithm based on the branch and bound (BB) method which partitions a matrix for any k, and we explore exact sparse matrix partitioning beyond bipartitioning. The algorithm has been implemented in a software package General Matrix Partitioner (GMP). We also present an integer linear programming (ILP) model for the same problem, based on a hypergraph formulation. We used both methods to determine optimal 2,3,4-way partitionings for a subset of small matrices from the SuiteSparse Matrix Collection. For k=2, BB outperforms ILP, whereas for larger k, ILP is superior. We used the results found by these exact methods for k=4 to analyse the performance of recursive bipartitioning (RB) with exact bipartitioning. For 46 matrices of the 89 matrices in our test set of matrices with less than 250 nonzeros, the communication volume determined by RB was optimal. For the other matrices, RB is able to find 4-way partitionings with communication volume close to the optimal volume.