Graph matching and MAP inference are essential problems in computer vision and machine learning. We introduce a novel algorithm that can accommodate both problems and solve them efficiently. Recent graph matching algorithms are based on a general quadratic programming formulation, that takes in consideration both unary and second-order terms reflecting the similarities in local appearance as well as in the pairwise geometric relationships between the matched features. In this case the problem is NP-hard and a lot of effort has been spent in finding efficiently approximate solutions by relaxing the constraints of the original problem. Most algorithms find optimal continuous solutions of the modified problem, ignoring during the optimization the original discrete constraints. The continuous solution is quickly binarized at the end, but very little attention is put into this final discretization step. In this paper we argue that the stage in which a discrete solution is found is crucial for good performance. We propose an efficient algorithm, with climbing and convergence properties, that optimizes in the discrete domain the quadratic score, and it gives excellent results either by itself or by starting from the solution returned by any graph matching algorithm. In practice it outperforms state-or-the art algorithms and it also significantly improves their performance if used in combination. When applied to MAP inference, the algorithm is a parallel extension of Iterated Conditional Modes (ICM) with climbing and convergence properties that make it a compelling alternative to the sequential ICM. In our experiments on MAP inference our algorithm proved its effectiveness by outperforming ICM and Max-Product Belief Propagation.