MIC2003: The Fifth Metaheuristics International Conference ???-1 Kyoto, Japan, August 25–28, 2003 Particle Swarm Optimization and Hill Climbing to Solve the Bandwidth Minimization Problem Andrew Lim a , Jing Lin b and Fei Xiao b a Dept of IEEM, HKUST, Clear Water Bay, Kowloon, Hong Kong b Dept of Comp. Sci, National University of Singapore, 3 Science Dr. 2, Singapore 117543 Email: iealim@ust.hk, {linjing,xiaofei}@comp.nus.edu.sg 1. Introduction The bandwidth minimization problem (BMP) is a well-known graph layout problem. It takes in a graph G = (V, E) as input, where V is the vertices set and E is the edge set, then outputs a permutation p of V that minimizes max (|p(i) – p(j)|) where (i, j) ∈ E (i.e. vertices i and j are connected). In the context of matrices, given a matrix A = [a ij ], the problem consists of finding a permutation of rows and columns that keeps all the nonzero elements of A in a band that is as close as possible to the main diagonal. The graph bandwidth minimization problem and the matrix bandwidth minimization problem can be transformed interchangeably by making A the incidence matrix of G. The BMP was first applied to speed up several computations on sparse matrices in fifties. For instance, in solving linear systems, Gaussian elimination can be performed in O(n b 2 ) on matrices of bandwidth b. This is a big win over the general O(n 3 ) algorithm if b << n. Other applications include problems in finite element methods for approximating solutions of partial differential equations, large-scale power transmissions systems, circuit design, hypertext layout, chemical kinetics and numerical geophysics. Due to its importance, the BMP has been studied extensively throughout the years. As the problem is NP-Complete, many heuristics have been developed to solve the BMP. The most well-known ones are the CutHill-McKee algorithm [1]. King’s algorithm [6], and the Gibbs-Poole-Stockmeyer algorithms (GPS) [3]. Most of them are based on a level structure of the graph, which partitions its vertices into levels L 0 , …, L n such that the endpoints of every edge in the graph are either in the same level L i or in two consecutive levels L i and L i+1 . By labeling the vertices level by level, a relatively good permutation giving a small bandwidth is obtained. Such algorithms based on a level structure are often grouped as the CM family. The most recent CM variant is the WBRA algorithm [2]. In the area of metaheuristics, Tabu search [2,7] and GRASP with path relinking [8] are applied to the BMP recently. They both outperform the level structure algorithms in terms of solution quality, in a reasonable amount of time, claiming to be the currently best-known algorithms to the BMP. In this paper, we propose a hybrid heuristic, Particle Swarm Optimization combined with Hill-Climbing (PSO-HC), to address the BMP. PSO-HC exploits a compact Particle Swarm Optimization (PSO) in order to generate high-quality renumbering which is then refined by an efficient implementation of the Hill Climbing local search heuristic. Computational experiments show that PSO-HC outperforms the newly proposed Tabu Search and GRASP with path relinking approaches in terms of solution quality, in reasonable time. We conclude this section by providing a table of notations used in the paper (Figure 1). The remaining of this paper is organized as follows: in the next section, we present the basic structure of our algorithm; in Section 3 and Section 4, we explain how PSO and Hill Climbing are applied to the BMP, respectively; in Section 5, we present computational result and compare