Summary: A typical first step of a direct solver for the linear system \(Mx=b\) is reordering of the symmetric matrix \(M\) to improve execution time and space requirements of the solution process. In this work, we propose a novel nested-dissection-based ordering approach that utilizes hypergraph partitioning.
Our approach is based on the formulation of graph partitioning by vertex separator (GPVS) problem as a hypergraph partitioning problem. This new formulation is immune to deficiency of GPVS in a multilevel framework and hence enables better orderings. In matrix terms, our method relies on the existence of a structural factorization of the input \(M\) matrix in the form of \(M=AA^T\) (or \(M=AD^2A^T)\).
We show that the partitioning of the row-net hypergraph representation of the rectangular matrix \(A\) induces a GPVS of the standard graph representation of matrix \(M\). In the absence of such factorization, we also propose simple, yet effective structural factorization techniques that are based on finding an edge clique cover of the standard graph representation of matrix \(M\), and hence applicable to any arbitrary symmetric matrix \(M\).
Our experimental evaluation has shown that the proposed method achieves better ordering in comparison to state-of-the-art graph-based ordering tools even for symmetric matrices where structural \(M=AA^T\) factorization is not provided as an input. For matrices coming from linear programming problems, our method enables even faster and better orderings.