Permuting Sparse Square Matrices Into Block Diagonal Form With Overlap

Abstract

In this whitepaper, we describe the problem of permuting sparse square matrices into block diagonal form with overlap (BDO)
and propose a graph partitioning algorithm for solving this problem. A block diagonal matrix with overlap is a block diagonal
matrix whose consecutive diagonal blocks may overlap. The objective in this permutation problem is to minimize the total
overlap size, whereas the permutation constraint is to maintain balance on the number of nonzeros in the diagonal blocks. This
permutation problem arises in the parallelization of an explicit formulation of multiplicative Schwarz preconditioner. We define
ordered Graph Partitioning by Vertex Separator (oGPVS) problem as an equivalent problem to this permutation problem. oGPVS
problem is a restricted version of Graph Partitioning by Vertex Separator (GPVS) problem and the aim is to find a partition of the
vertices into K ordered vertex parts and K-1 ordered separators where each two consecutive parts can be connected through only
a separator, a separator can only connect two consecutive parts, and each two consecutive separators can be adjacent. The
objective in the oGPVS problem is to minimize the total number of vertices in the separators, whereas the partitioning objective
is to maintain balance on the part weights where part weight is defined as the sum of the weights of vertices in that part. To solve
oGPVS problem, we utilized recursive bipartitioning paradigm and fixed vertices in our proposed oGPVS algorithm. We tested
the performance of our algorithm in a wide range of matrices in comparison to another graph partitioning algorithm that solves
the same problem. Results showed that the oGPVS algorithm performs better than the other algorithm in terms of overlap size.