Flow metrics

Abstract

We introduce flow metrics as a relaxation of path metrics (i.e. linear orderings). They are defined by polynomial-sized linear programs and have interesting properties including spreading. We use them to obtain relaxations for several NP-hard linear ordering problems such as the minimum linear arrangement and minimum pathwidth. Our approach has the advantage of achieving the best-known approximation guarantees for these problems using the same relaxation and essentially the same rounding for all the problems and varying only the objective function from problem to problem. This is in contrast to the current state of the literature where each problem warrants either a new relaxation or a new rounding or both. We also characterize a natural projection of the relaxation.