Approximating the Bandwidth via Volume Respecting Embeddings

A linear arrangement of an n-vertex graph is a one-to-one mapping of its vertices to the integers f1; : : : ; ng. The bandwidth of a linear arrangement is the maximum diierence between mapped values of adjacent vertices. The problem of nding a linear arrangement with smallest possible bandwidth in NP-hard. We present a random-ized algorithm that runs in nearly linear time and outputs a linear arrangement whose bandwidth is within a polylogarithmic multiplicative factor of optimal. Our algorithm is based on a new notion, called volume respecting embeddings, which is a natural extension of small distortion embeddings of Bourgain and of Linial, London and Rabinovich.