Abstract Many computer vision problems require optimization of binary non-submodular energies. In this context, local iterative submodularization techniques based on trust region (LSA-TR) and auxiliary functions (LSA-AUX) have been recently proposed [9]. They achieve state-of-the-art-results on a number of computer vision applications. In this paper we extend the LSA-AUX framework in two directions. First, unlike LSA-AUX, which selects auxiliary functions based solely on the current solution, we propose to incorporate several additional criteria. This results in tighter bounds for confifigurations that are more likely or closer to the current solution. Second, we propose move-making extensions of LSA-AUX which achieve tighter bounds by restricting the search space. Finally, we evaluate our methods on several applications. We show that for each application at least one of our extensions signifificantly outperforms the original LSA-AUX. Moreover, the best extension of LSA-AUX is comparable to or better than LSA-TR on four out of six applications