Reducing the bandwidth of a sparse matrix with tabu search

Abstract

. The problem of reducing the bandwidth of a matrix consists of finding a permutation of the rows and columns that keeps the nonzero elements in a band that is as close as possible to the main diagonal of the matrix. This NP-complete problem can also be formulated as a labeling of vertices on a graph, where edges are the nonzero elements of the corresponding symmetrical matrix. Many bandwidth reduction algorithms have been developed since the 1960s and applied to structural engineering, fluid dynamics and network analysis. For the most part, these procedures do not incorporate metaheuristic elements, which is one of the main goals of our current development. Another equally important goal is to design and test a special type of candidate list strategy and a move mechanism to be embedded in a tabu search procedure for the bandwidth reduction problem. This candidate list strategy accelerates the selection of a move in the neighborhood of the current solution in any given iteration. Our extensive experimentation shows that the proposed procedure outperforms the best-known algorithms in terms of solution quality consuming a reasonable computational effort.

A branch and bound algorithm for the matrix bandwidth minimization

“…We have considered the set of 113 instances from the HarwellBoeing Sparse Matrix Collection, introduced in Martí et al (2001) for this problem and used subsequently in different papers. We have also included 40 random instances in our experiments.…”

“…For many years researchers were only interested in designing relatively simple heuristic procedures and sacrificed solution quality for speed (see for example Cuthill andMcKee 1969, or Gibbs, Poole andStockmeyer 1976). Recently, metaheuristics such as tabu search (Martí et al 2001) or GRASP (Piñana et al 2004), have been developed for this problem in order to obtain high quality solutions. The second line of research consists of the development of theoretical results (mainly lower bounds) and exact methods for the MBMP.…”

Context-Independent Scatter and Tabu Search for Permutation Problems

“…It should be mentioned that in the case of LOP and TSP the best solutions considered are the optimal solutions as given in the public libraries. In the case of the BRP the best solutions are from Martí et al (2000) and the best solutions for the SMS instances are due to Laguna et al (1993).…”

A Hyper-Heuristic Approach to Evolving Algorithms for Bandwidth Reduction Based on Genetic Programming

“…For example, Tabu search was employed by Marti et al [30], while Lim et al [27] used a hybrid between genetic algorithm and hill-climbing to solve this problem. Lim et al also introduced two other hybrid algorithms to solve the BMP: one combining ant colony optimization with hill-climbing [25] and one combining particle swarm optimization with hillclimbing [26].…”