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This paper presents a class of power grid analysis and optimization techniques, all of which are based on the algebraic-multigrid (AMG) method. First, a new AMG-based reduction scheme is proposed to improve the efficiency of reducing the problem size for power grid analysis and optimization. Next, with the proposed reduction technique, a fast transient-analysis method is developed and extended to an accurate solver with error control mechanism. After that, the scope of this method is further broadened for handling the analysis of the modified grid. Finally, a fast decap-allocation (DA) scheme based on AMG is suggested. Experimental results show that these techniques not only achieve a significant speedup over reported industrial methods but also enhance the quality of solutions. By using the proposed techniques, transient analysis with 200 time steps on a 1.6-M-node power grid can be completed in less than 5 min; dc analysis on the same circuit can reach an accuracy of in about 141 s. Our DA can process a circuit with up to one million nodes in about 11 min.