An Analysis of Spectral Envelope Reduction via Quadratic Assignment Problems

A new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size was described in [Barnard, Pothen, and Simon, Numer. Linear Algebra Appl., 2 (1995), pp. 317--334]. The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. In this paper we provide an analysis of the spectral envelope reduction algorithm. We describe related 1- and 2-sum problems; the former is related to the envelope size, while the latter is related to an upper bound on the work in an envelope Cholesky factorization. We formulate these two problems as quadratic assignment problems and then study the 2-sum problem in more detail. We obtain lower bounds on the 2-sum by considering a relaxation of the problem and then show that the spectral ordering finds a permutation matrix closest to an orthogonal matrix attaining the lower bound. This provides a stronger justification of the spectral envelope reduction algorithm than previously known. The lower bound on the 2-sum is seen to be tight for reasonably "uniform" finite element meshes. We show that problems with bounded separator sizes also have bounded envelope parameters.