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Motivated by current trends in cloud computing, we study a version of the generalized assignment problem where a set of virtual processors has to be implemented by a set of identical processors. For literature consistency, we say that a set of virtual machines (VMs) is assigned to a set of physical machines (PMs). The optimization criterion is to minimize the power consumed by all the PMs. We term the problem Virtual Machine Assignment (VMA). Crucial differences with previous work include a variable number of PMs, that each VM must be assigned to exactly one PM (i.e., VMs cannot be implemented fractionally), and a minimum power consumption for each active PM. Such infrastructure may be strictly constrained in the number of PMs or in the PMs' capacity, depending on how costly (in terms of power consumption) it is to add a new PM to the system or to heavily load some of the existing PMs. Low usage or ample budget yields models where PM capacity and/or the number of PMs may be assumed unbounded for all practical purposes. We study four VMA problems depending on whether the capacity or the number of PMs is bounded or not. Specifically, we study hardness and online competitiveness for a variety of cases. To the best of our knowledge, this is the first comprehensive study of the VMA problem for this cost function. (c) 2015 Elsevier B.V. All rights reserved.