D-dimensional Arrangement Revisited

We revisit the d-dimensional arrangement problem and analyze the performance ratios of previously proposed algorithms based on the linear arrangement problem with d-dimensional cost. The two problems are related via space-filling curves and recursive balanced bipartitioning. We prove that the worst-case ratio of the optimum solutions of these problems is Θ(log n), where n is the number of vertices of the graph. This invalidates two previously published proofs of approximation ratios for d-dimensional arrangement. Furthermore, we conclude that the currently best known approximation ratio for this problem is O(log n).