On the use of suboptimal matchings for scaling and ordering sparse symmetric matrices

The use of matchings is a powerful technique for scaling and ordering sparse matrices prior to the solution of a linear system Ax D b. Traditional methods such as implemented by the HSL software package MC64 use the Hungarian algorithm to solve the maximum weight maximum cardinality matching problem. However, with advances in the algorithms and hardware used by direct methods for the parallelization of the factorization and solve phases, the serial Hungarian algorithm can represent an unacceptably large proportion of the total solution time for such solvers. Recently, auction algorithms and approximation algorithms have been suggested as alternatives for achieving near-optimal solutions for the maximum weight maximum cardinality matching problem. In this paper, the efficacy of auction and approximation algorithms as replacements for the Hungarian algorithm is assessed in the context of sparse symmetric direct solvers when used in problems arising from a range of practical applications. High-cardinality suboptimal matchings are shown to be as effective as optimal matchings for the purposes of scaling. However, matching-based ordering techniques require that matchings are much closer to optimality before they become effective. The auction algorithm is demonstrated to be capable of finding such matchings significantly faster than the Hungarian algorithm, but our 1 2 -approximation matching approach fails to consistently achieve a sufficient cardinality. Copyright © 2015 The Authors Numerical Linear Algebra with Applications Published by John Wiley & Sons Ltd.