Hardness results for approximating the bandwidth

The bandwidth of an n-vertex graph G is the minimum value b such that the vertices of G can be mapped to distinct integer points on a line without any edge being stretched to a distance more than b. Previous to the work reported here, it was known that it is NP-hard to approximate the bandwidth within a factor better than 3/2. We improve over this result in several respects. For certain classes of graphs (such as cycles of cliques) for which it is easy to approximate the bandwidth within a factor of 2, we show that approximating the bandwidth within a ratio better than 2 is NP-hard. For caterpillars (trees in which all vertices of degree larger than two lie on one path) we show that it is NP-hard to approximate the bandwidth within any constant, and that an approximation ratio of will imply a quasi-polynomial time algorithm for NP (when c is a sufficiently small constant).