Graphical models such as Markov random fields have been successfully applied to a wide variety of fields, from computer vision and natural language processing, to computational biology. Exact probabilistic inference is generally intractable in complex models having many dependencies between the variables. We present new approaches to approximate inference based on linear programming (LP) relaxations. Our algorithms optimize over the cycle relaxation of the marginal polytope, which we show to be closely related to the first lifting of the Sherali-Adams hierarchy, and is significantly tighter than the pairwise LP relaxation. We show how to efficiently optimize over the cycle relaxation using a cutting-plane algorithm that iteratively introduces constraints into the relaxation. We provide a criterion to determine which constraints would be most helpful in tightening the relaxation, and give efficient algorithms for solving the search problem of finding the best cycle constraint to add according to this criterion. By solving the LP relaxations in the dual, we obtain efficient message-passing algorithms that, when the relaxations are tight, can provably find the most likely (MAP) configuration. Our algorithms succeed at finding the MAP configuration in protein side-chain placement, protein design, and stereo vision problems.