A large part of image analysis is about breaking things into pieces. Decompositions of a graph are a mathematical abstraction of the possible outcomes. This talk is about optimization problems whose feasible solutions define decompositions of a graph. One example is the correlation clustering problem whose feasible solutions relate one-to-one to the decompositions of a graph, and whose objective function puts a cost or reward on neighboring nodes ending up in distinct components. This talk shows applications of this problem and proposed generalizations to diverse image analysis tasks. It sketches algorithms for finding feasible solutions for large instances in practice, solutions that are often superior in the metrics of application-specific benchmarks. It also sketches algorithms for finding lower bounds and points to new findings and open problems of polyhedral geometry in this context.