Model Fitting and Fast Energy Minimization with Label Cost Abstract: The $ \alpha $-expansion algorithm has had a significant impact in computer vision due to its generality, effectiveness, and speed. Thus far it can only minimize energies that involve unary, pairwise, and specialized higher-order terms. We show how to extend α-expansion so that it can simultaneously optimize label costs as well. An energy with label costs can penalize a solution based on the set of labels that appear in it. The simplest special case is to penalize the number of labels in the solution. Our energy is quite general, and we prove optimality bounds for our algorithm. A natural application of label costs is multi-model fitting, which we discuss in detail. We demonstrate many applications in vision: homography detection, motion segmentation, and unsupervised image segmentation. Our C++/MATLAB implementation is publicly available.