Exact and Heuristic Solutions to the Bandwidth Minimization Problem

Abstract

The bandwidth minimization problem is a classical combinatorial optimization problem studied since about 1960. It is formulated as follows. Given a connected graph G=(V,E) with n vertices, the task is to find a permutation l of the vertices (also called a labeling), i.e., a bijection between V and {1,2,...,n}, such that the maximum difference |l(u)-l(v)|, for uv in E, is minimized. This problem is NP-hard, even for binary trees. Applications of the bandwidth problem can be found in many areas: solving systems of linear equations, data storage, electronic circuit design, and recently in compression of topological information from digital road networks. In this dissertation we report our contributions of both heuristic and exact methods for the bandwidth problem. On the heuristic side, we start by modifying a heuristic method which exploits properties of the graph. Next we propose an approximate objective function for the bandwidth problem. It is very sensitive to alterations in a permutation and can thus be used efficiently in global optimization heuristic methods. A simulated annealing method using the approximate objective function is reported. We also present an application of the bandwidth problem to the compression of topological information of digital road networks. For exact methods, which are our main focus, we formulate the concept of a partial permutation. Based on this concept, we introduce new constraints for the bandwidth problem and apply them efficiently in a branch-and-bound algorithm. We analyze the relation between certain partial permutations and show that some partial permutations are dominated by others. Therefore, they can be eliminated in the branch-and-bound tree and this reduces the search space and running time. Furthermore, we enhance the use of partial permutations in branch-and-bound algorithms with a 2-labeling scheme, supported by the dominance rule. Instead of extending the partial permutation one-by-one, our scheme uses two vertices simultaneously. We evaluate our algorithms on a popular benchmark suite which comprises 113 instances with less than 1,000 vertices each. In many cases our work improves on the best known lower bound in the literature. Moreover, our exact algorithms are capable of computing lower bounds for much larger instances. We perform computational experiments on a second suite of 36 instances with more than 1,000 vertices each, whose best known lower bound so far is the generic theoretical one. We can improve this bound for some instances in this suite, the largest such instance having about 15,600 vertices. Finally, we parallelize our branch-and-bound algorithms and run the solver on a parallel cluster with 256 processors, improving the lower bound for some instances in the first benchmark suite even further. %%%%Das Bandwidth Minimization Problem ist ein klassisches kombinatorisches Optimierungsproblem, das bereits seit etwa 1960 untersucht wird: Zu einem gegebenen zusammenhangenden Graphen G=(V,E) der Ordnung n ist eine Permutation l der Knoten (auch Labeling…