Algebraic multigrid (AMG) is an important method for solving the large sparse linear systems that arise in many PDE-based scientific simulation codes.  A major component of algebraic multigrid methods is the selection of coarse grids and the construction of interpolation.  The idea of compatible relaxation (CR) was introduced by Brandt as a cheap method for measuring the quality of a coarse grid.  In this talk, we will review the theory behind CR, describe our CR-based coarsening algorithm, and discuss aspects of the method that require additional development such as coarsening for systems of PDEs.  We will also discuss CR's ability to predict the convergence behavior of the AMG method and ideas for improving the accuracy of its predictions.  Finally, we will talk about issues of parallelizing these methods to run on massively parallel computers.