We propose an efficient and adaptive method for MAP-MRF inference that provides increasingly tighter upper and lower bounds on the optimal objective. Our method starts by solving the first-order LOCAL(G) linear programming relaxation. This is followed by an adaptive tightening of the relaxation where we incrementally add higher-order interactions to enforce proper marginalization over groups of variables. Computing the best interaction to add is an NP-hard problem. We show good solutions to this problem can be readily obtained from “local primal-dual gaps ” given the current primal solution and a dual reparameterization vector. This is not only extremely efficient, but in contrast to previous approaches, also allows us to search over prohibitively large sets of candidate interactions to add. We demonstrate the superiority of our approach on MAP-MRF inference problems encountered in computer vision.