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This paper presents important, new results of a study on the problem of task scheduling and voltage allocation in dynamically variable voltage processors, the purpose of which was minimization of processor energy consumption. The contributions are twofold: (1) For given multiple discrete supply voltages and tasks with arbitrary arrival-time/deadline constraints, we propose a voltage allocation technique that produces a feasible task schedule with optimal processor energy consumption. (2) We then extend the problem to include the case in which tasks have nonuniform loads (i.e.; switched) capacitances and solve it optimally. The proposed technique, called Alloc-vt, in (1) is based on the prior results in [Yao, Demers and Shenker. 1995. In Proceedings of IEEE Symposium on Foundations of Computer Science. 374--382] (which is optimal for dynamically continuously variable voltages, but not for discrete ones) and [Ishihara and Yasuura. 1998. In Proceedings of International Symposium on Low Power Electronics and Design. 197--202] (which is optimal for a single task, but not for multiple tasks), whereas the proposed technique, called Alloc-vtcap, in (2) is based on an efficient linear programming (LP) formulation. Both techniques solve the allocation problems optimally in polynomial time.