Ant colony optimization with hill climbing for the bandwidth minimization problem

2014

Comput Optim Appl

Large sparse symmetric matrix problems arise in a number of scientific and engineering fields such as fluid mechanics, structural engineering, finite element analysis and network analysis. In all such problems, the performance of solvers depends critically on the sum of the row bandwidths of the matrix, a quantity known as envelope size. This can be reduced by appropriately reordering the rows and columns of the matrix, but for an N × N matrix, there are N ! such permutations, and it is difficult to predict how each permutation affects the envelope size without actually performing the reordering of rows and columns. These two facts compounded with the large values of N used in practical applications, make the problem of minimising the envelope size of a matrix an exceptionally hard one. Several methods have been developed to reduce the envelope size. These methods are mainly heuristic in nature and based on graph-theoretic concepts. While metaheuristic approaches are popular 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 size of sparse matrices is presented and evaluated against four of the best-known and broadly used envelope reduction algorithms. The results obtained on a wide-ranging set of standard benchmarks from the Harwell-Boeing sparse matrix collection show that the proposed method compares very favourably with these algorithms.

Section: Relevant Existing Algorithmsmentioning

confidence: 96%

Addressing the envelope reduction of sparse matrices using a genetic programming system

2014

Comput Optim Appl

“…We will also see if performance can be further enhanced by mutating constants, by using non-random populations and by hybridising the system by incorporating a local optimiser as was done in [12][13][14]. In addition, we will need to assess the generality and scalability of the proposed method by testing it with larger datasets and datasets including larger matrices.…”

Section: Discussionmentioning

confidence: 99%

“…For example, Tabu search was employed by Marti et al [17] while Lim et al [14,16] used a hybrid between genetic algorithms and hill-climbing to solve this problem. Lim et al also introduced two other hybrid algorithms to solve BMP: one combining ant colony optimization with hill-climbing [12] and one combining particle swarm optimization with hillclimbing [13]. Recently also simulated annealing has been used to attack the problem [21].…”

Section: Introductionmentioning

confidence: 99%

A Genetic Programming Approach to the Matrix Bandwidth-Minimization Problem

Parallel Problem Solving From Nature, PPSN XI

2010

Abstract. The bandwidth of a sparse matrix is the distance from the main diagonal beyond which all elements of the matrix are zero. The bandwidth minimisation problem for a matrix consists of finding the permutation of rows and columns of the matrix which ensures that the non-zero elements are located in as narrow a band as possible along the main diagonal. This problem, which is known to be NP-complete, can also be formulated as a vertex labelling problem for a graph whose edges represent the non-zero elements of the matrix. In this paper, a Genetic Programming approach is proposed and tested against two of the best-known and widely used bandwidth reduction algorithms. Results have been extremely encouraging.Keywords: Bandwidth Minimization Problem; Genetic Programming; Graph Labelling; Sparse Matrices; Combinatorial Optimisation. BackgroundThe Bandwidth Minimization Problem (BMP) is a very well-known problem, familiar to applied mathematicians and arising in many applications in science and engineering [18]. BMP consists of finding the permutation of rows and columns of a matrix which ensures that the non-zero elements are located in as narrow a band as possible along the main diagonal. One of the most common applications of bandwidth-minimisation algorithms arises from the need to efficiently solve large systems of equations [19]. In such a scenario, more efficient solutions are obtained if the rows and columns of the matrix representing the set of equations can be permuted in such a way that the bandwidth of the matrix is minimized [19]. BMP has also connections with a wide range of other problems, including: finite element analysis of mechanical systems, large scale power transmission systems, circuit design, VLSI design, data storage, chemical kinetics, network survivability, numerical geophysics, industrial electromagnetics, saving large hypertext media and topology compression of road networks.The BMP is NP-complete [18] and, hence, it is highly unlikely that there exists an algorithm which finds the minimum bandwidth of a matrix in polynomial time. It has also been proved that the BMP is NP-complete even for trees with

“…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]. More recently, simulated annealing has also been used to attack the problem [41].…”

Section: Introductionmentioning

confidence: 99%

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

Research and Development in Intelligent Systems XXVIII

2011

The bandwidth reduction problem is a well-known NP-complete graphlayout problem that consists of labeling the vertices of a graph with integer labels in such a way as to minimize the maximum absolute difference between the labels of adjacent vertices. The problem is isomorphic to the important problem of reordering the rows and columns of a symmetric matrix so that its non-zero entries are maximally close to the main diagonal -a problem which presents itself in a large number of domains in science and engineering. A considerable number of methods have been developed to reduce the bandwidth, among which graph-theoretic approaches are typically faster and more effective. In this paper, a hyper-heuristic approach based on genetic programming is presented for evolving graph-theoretic bandwidth reduction algorithms. The algorithms generated from our hyper-heuristic are extremely effective. We test the best of such evolved algorithms on a large set of standard benchmarks from the Harwell-Boeing sparse matrix collection against two state-of-the-art algorithms from the literature. Our algorithm outperforms both algorithms by a significant margin, clearly indicating the promise of the approach.