On approximate non-submodular minimization via tree-structured supermodularity

We address the problem of minimizing non-submodular functions where the supermodularity is restricted to tree-structured pair-wise terms. We are motivated by several real world applications, which require submodu-larity along with structured supermodular-ity, and this forms a rich class of expressive models, where the non-submodularity is restricted to a tree. While this problem is NP hard (as we show), we develop several practical algorithms to find approximate and near-optimal solutions for this problem, some of which provide lower and others of which provide upper bounds thereby allowing us to compute a tightness gap. We also show that some of our algorithms can be extended to handle more general forms of supermodular-ity restricted to arbitrary pairwise terms. We compare our algorithms on synthetic data, and also demonstrate the advantage of the formulation on the real world application of image segmentation, where we incorporate structured supermodularity into higher-order submodular energy minimization.