On the application of Genetic Programming to the envelope reduction problem

Large sparse matrices characterise the linear systems found in various scientific and engineering domains such as fluid mechanics, structural engineering, finite element analysis and network analysis. The ordering of the rows and columns of a matrix determines how close to the main diagonal its non-zero elements are, which in turn greatly influences the performance of solvers for the associated linear system. The reduction of the sum of the distance of non-zero elements from the matrix's main diagonal - a quantity known as envelope - is thus a key issue in many domains. Formally, the problem consists in finding a permutation of the rows and columns of a matrix which minimises its envelope. The problem is known to be NP-complete. A considerable number of methods have been proposed for reducing the envelope. These methods are mostly based on graph-theoretic concepts. While metaheuristic approaches are viable alternatives to classical optimisation techniques in a variety of domains, in the case of the envelope reduction problem, there has been a very limited exploration of such methods. In this paper, a Genetic Programming system capable of reducing the envelope of sparse matrices is presented. We evaluate our method on a set of standard benchmarks from the Harwell-Boeing sparse matrix collection against four state-of-the-art algorithms from the literature. The results obtained show that the proposed method compares very favourably with these algorithms.