Hypergraph Partitioning-Based Fill-Reducing Ordering for Symmetric Matrices

TL;DR: A novel nested-dissection-based ordering approach based on the formulation of graph partitioning by vertex separator (GPVS) problem as a hypergraph partitioning problem that enables better orderings of matrices coming from linear programming problems.

Abstract: 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.