Parallel Computation for the Bandwidth Minimization Problem

Abstract

The bandwidth minimization problem is a classical combinatorial optimization problem that has been studied since around 1960. It is formulated as follows. Given a connected graph G = (V;E) with n nodes and m edges, find a labeling π, i.e. a bijection between V and {1; 2;… ; n}, such that the maximum difference |π(u)–(v)|, uv e E, is minimized. This problem is NP-hard even for binary trees [3]. Though much work on the bandwidth problem has been done, the gaps between best known lower and upper bounds for benchmark problems is still large. Applications of the bandwidth problem can be found in many areas: solving systems of linear equations, data storing, electronic circuit design, and recently in topology compression of road networks [7]. Due to the hardness of bandwidth minimization, much research has dealt with heuristic methods. These range from structured methods to metaheuristic methods, genetic algorithm and scatter search. The reader can refer to [1] for detailed references in this areas. For solving the problem to optimality, dynamic programming and branch-andbound have been used but with little success for sparse graphs. Caprara and Salazar [2] have proposed new lower bounds and strong integer linear programming (ILP) formulations to compute the bounds more effectively. Their method is therefore used in our parallel implementation documented in this paper. We will report about computational results for problem instances with up to 200 nodes.