This paper proposes a framework that provides significant speed-ups and also improves the effectiveness of general message passing algorithms based on dual LP relaxations. It is applicable to both pairwise and higher order MRFs, as well as to any type of dual relaxation. It relies on combining two ideas. The first one is inspired by algebraic multigrid approaches for linear systems, while the second one employs a novel decimation strategy that carefully fixes the labels for a growing subset of nodes during the course of a dual LP-based algorithm. Experimental results on a wide variety of vision problems demonstrate the great effectiveness of this framework.