A Fast Algorithm for Finding an Optimal Ordering for Vertex Elimination on a Graph

Abstract

This paper gives a graph-theoretic approach to the problem of finding an optimal ordering for Gaussian elimination on a sparse matrix. A set of new “fill-ins” produced by Gaussian elimination on a matrix A is characterized by a triangulation induced by vertex elimination on a graph associated with A. A triangulation T is minimal (minimum) if there exists no triangulation $\hat T$ such that $\hat{T} \subset \subset T(|\hat{T}| < |T|)$, where $ \subset \subset $ denotes the strict inclusion, and an ordering $\alpha $ is optimal (optimum) if a minimal (minimum) triangulation is generated by a. An optimum ordering is necessarily optimal but not conversely. An efficient algorithm for finding an optimal ordering in $O(M \cdot N)$ time is presented, where M is the number of vertices and N is the number of edges of the graph being considered.