Rounding Guarantees for Message-Passing MAP Inference with Logical Dependencies

We present the equivalence of the first-order local consistency relaxation and the MAX SAT relaxation of Goemans and Williamson [1] for a class of MRFs we refer to as logical MRFs. This allows us to combine the advantages of both approaches into a single technique: solving the local consistency relaxation with any of a number of message-passing algorithms, and then improving the solution quality via a guaranteed rounding procedure when the relaxation is not tight. Logical MRFs are a general class of models that can incorporate many common dependencies, such as mixtures of submodular and supermodular potentials, and logical implications. They can be used for many tasks, including natural language processing, computer vision, and computational social science.