Reordering sparse matrices into block-diagonal column-overlapped form

Abstract

Many scientific and engineering applications necessitate computing the minimum norm solution of a sparse underdetermined linear system of equations. The minimum 2-norm solution of such systems can be obtained by a recent parallel algorithm, whose numerical effectiveness and parallel scalability are validated in both shared- and distributed-memory architectures. This parallel algorithm assumes the coefficient matrix in a block-diagonal column-overlapped (BDCO) form, which is a variant of the block-diagonal form where the successive diagonal blocks may overlap along their columns. The total overlap size of the BDCO form is an important metric in the performance of the subject parallel algorithm since it determines the size of the reduced system, solution of which is a bottleneck operation in the parallel algorithm. In this work, we propose a hypergraph partitioning model for reordering sparse matrices into BDCO form with the objective of minimizing the total overlap size and the constraint of maintaining balance on the number of nonzeros of the diagonal blocks. Our model makes use of existing partitioning tools that support fixed vertices in the recursive bipartitioning paradigm. Experimental results validate the use of our model as it achieves small overlap size and balanced diagonal blocks.