What Energy Functions can be Minimized via Graph Cuts?

Vladimir Kolmogorov and Ramin Zabih.

In IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 26(2):147-159, February 2004.
Earlier version appeared in European Conference on Computer Vision (ECCV), May 2002 (best paper award).


In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions, and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a general-purpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible, and then follow our construction to create the appropriate graph.



Note: After the paper has been published, I found out that most of its results appeared a long time ago. The fact that submodular functions can be minimized via graph cuts (min-cut/max-flow algorithm) is well-known in the optimization literature, see e.g. Graph construction for submodular functions with triple cliques was given in For a review of pseudo-boolean optimization literature, see