This paper presents a new adaptive graph-cut based move-making algorithm for energy minimization. Traditional move-making algorithms such as Expansion and Swap operate by searching for better solutions in some predefined moves spaces around the current solution. In contrast, our algorithm uses the primal-dual interpretation of the Expansion-move algorithm to adaptively compute the best move-space to search over. At each step, it tries to greedily find the move-space that will lead to biggest decrease in the primal-dual gap. We test different variants of our algorithm on a variety of image labelling problems such as object segmentation and stereo. Experimental results show that our adaptive strategy significantly outperforms the conventional Expansion-move algorithm, in some cases cutting the runtime by 50%.