C.: Global MAP-optimality by shrinking the combinatorial search area with convex relaxation

We consider energy minimization for undirected graphical models, also known as the MAP-inference problem for Markov random fields. Although combinatorial methods, which return a provably optimal integral solution of the problem, made a significant progress in the past decade, they are still typically unable to cope with large-scale datasets. On the other hand, large scale datasets are often defined on sparse graphs and convex relaxation methods, such as linear programming relax-ations then provide good approximations to integral solutions. We propose a novel method of combining combinatorial and convex program-ming techniques to obtain a global solution of the initial combinatorial problem. Based on the information obtained from the solution of the convex relaxation, our method confines application of the combinatorial solver to a small fraction of the initial graphical model, which allows to optimally solve much larger problems. We demonstrate the efficacy of our approach on a computer vision energy mini-mization benchmark.