We consider energy minimization for undirected graphical models, also known as MAP-inference problem for Markov random fields. Although combinatorial methods, which return a provably optimal integral solution of the problem, made a big progress in the past decade, they are still typically unable to cope with large-scale datasets. On the other hand, large scale datasets are typically defined on sparse graphs, and convex relaxation methods, such as linear programming relaxations often provide good approximations to integral solutions. We propose a novel method of combining combinatorial and convex programming 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 big problems. We demonstrate the power of our approach on a computer vision energy minimization benchmark.