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In this paper, we present novel techniques that improve the computational and memory efficiency of algorithms for solving multi-label energy functions arising from discrete MRFs or CRFs. These methods are motivated by the observations that the performance of minimization algorithms depends on: 1) the initialization used for the primal and dual variables and 2) the number of primal variables involved in the energy function. Our first method (dynamic α-expansion) works by "recycling" results from previous problem instances. The second method simplifies the energy function by "reducing" the number of unknown variables present in the problem. Further, we show that it can also be used to generate a good initialization for the dynamic α-expansion algorithm by "reusing" dual variables. We test the performance of our methods on energy functions encountered in the problems of stereo matching and color and object-based segmentation. Experimental results show that our methods achieve a substantial improvement in the performance of α-expansion, as well as other popular algorithms such as sequential tree-re-weighted message passing and max-product belief propagation. We also demonstrate the applicability of our schemes for certain higher order energy functions, such as the one described, for interactive texture-based image and video segmentation. In most cases, we achieve a 10-15 times speed-up in the computation time. Our modified α-expansion algorithm provides similar performance to Fast-PD, but is conceptually much simpler. Both α-expansion and Fast-PD can be made orders of magnitude faster when used in conjunction with the "reduce" scheme proposed in this paper.