Discrete Optimization Variable neighbourhood search for bandwidth reduction Nenad Mladenovic a, * , Dragan Urosevic b , Dionisio Pérez-Brito c , Carlos G. García-González d a School of Mathematics, Brunel University – West London, UK b Mathematical Institute SANU Belgrade, Serbia c Dpto. de Estadística, Investigación Operativa y Computación, Universidad de La Laguna, Spain d Dpto. de Economía de las Instituciones, Estadística Económica y Econometría, Universidad de La Laguna, Spain article info Article history: Received 23 March 2008 Accepted 9 December 2008 Available online 24 December 2008 Keywords: Combinatorial optimization Matrix bandwidth minimization Metaheuristics Variable neighbourhood search abstract The problem of reducing the bandwidth of a matrix consists of finding a permutation of rows and col- umns of a given matrix which keeps the non-zero elements in a band as close as possible to the main diagonal. This NP-complete problem can also be formulated as a vertex labelling problem on a graph, where each edge represents a non-zero element of the matrix. We propose a variable neighbourhood search based heuristic for reducing the bandwidth of a matrix which successfully combines several recent ideas from the literature. Empirical results for an often used collection of 113 benchmark instances indi- cate that the proposed heuristic compares favourably to all previous methods. Moreover, with our approach, we improve best solutions in 50% of instances of large benchmark tests. Crown Copyright Ó 2008 Published by Elsevier B.V. All rights reserved. 1. Introduction Let GðV ; EÞ be a graph with node set V and edge set E and let f : V !f1; ... ; ng be a labelling (or colouring) of nodes, where n ¼jV j. Let us define the bandwidth of a graph G under f as Bðf Þ¼ maxfjf ðuÞ f ðv Þj 8ðu; v Þ2 Eg: Moreover, if we define the bandwidth of vertex v under f as Bðf ; v Þ¼ max u:ðu;v Þ2E jf ðuÞ f ðv Þj; ð1Þ the bandwidth may also be expressed as Bðf Þ¼ max v Bðf ; v Þ. Then the bandwidth minimization problem is to find a labelling f  , which min- imizes Bðf Þ, i.e. B  ¼ Bðf  Þ¼ min f Bðf Þ¼ min f max ðu;v Þ2E jf ðuÞ f ðv Þj and the aim of bandwidth reduction is to find a labelling which gives an approximate solution to this problem. If we represent G by its adjacency matrix A, the problem can be formulated as finding a permutation of the rows and the columns of A which keeps all the non-zero elements in a band as close as possible to the main diagonal. This is why this problem is known as the Matrix bandwidth minimization problem (MBMP for short), where bandwidth is defined as the maximum distance of a nonzero element from the main diagonal. Example. Given a graph G with n ¼ 5 and the given labelling f : f ðv 1 Þ¼ 4; f ðv 2 Þ¼ 2; f ðv 3 Þ¼ 3; f ðv 4 Þ¼ 1; f ðv 5 Þ¼ 5. As indicated in Fig. 1, the bandwidth of G under the labelling f is Bðf Þ¼ 4 and the edge which produces this value is the one which is rounded. The adjacency matrix of the graph under labelling f is as follows: 0377-2217/$ - see front matter Crown Copyright Ó 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.12.015 * Corresponding author. E-mail addresses: Nenad.Mladenovic@brunel.ac.uk (N. Mladenovic), draganu@turing.mi.sanu.ac.yu (D. Urosevic), Dperez@ull.es (D. Pérez-Brito), Cggarcia@ull.es (C.G. García-González). European Journal of Operational Research 200 (2010) 14–27 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor