Eﬃcient global optimization techniques such as graph cut exist for energies corresponding to binary image segmentation from low-level cues. However, introducing a high-level prior such as a shape prior or a color-distribution prior into the segmentation process typically results in an energy that is much harder to optimize. The main contribution of the paper is a new global optimization framework for a wide class of such energies. The framework is built upon two powerful techniques: graph cut and branch-and-bound. These techniques are uniﬁed through the derivation of lower bounds on the energies. Being computable via graph cut, these bounds are used to prune branches within a branch-and-bound search.