Maximum a posteriori (MAP) inference in graphical models requires that we maximize the sum of two terms: a data-dependent term, encoding the conditional likelihood of a certain labeling given an observation, and a data-independent term, encoding some prior on labelings. Often, data-dependent factors contain fewer latent variables than dataindependent factors -for instance, many grid and tree-structured models contain only firstorder conditionals despite having pairwise priors. In this paper, we

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... note that MAPinference in such models can be made substantially faster by appropriately preprocessing their data-independent terms. Our main result is to show that message-passing in any such pairwise model has an expected-case exponent of only 1.5 on the number of states per node, leading to significant improvements over existing quadratic-time solutions.