Graph cuts-based algorithms have achieved great success in energy minimization for many computer vision applications. These algorithms provide approximated solutions for multi-label energy functions via move-making approach. This approach fuses the current solution with a proposal to generate a lower-energy solution. Thus, generating the appropriate proposals is necessary for the success of the move-making approach. However, not much research efforts has been done on the generation of "good" proposals, especially for non-metric energy functions. In this paper, we propose an application-independent and energy-based approach to generate "good" proposals. With these proposals, we present a graph cuts-based move-making algorithm called GA-fusion (fusion with graph approximation-based proposals). Extensive experiments support that our proposal generation is effective across different classes of energy functions. The proposed algorithm outperforms others both on real and synthetic problems.