Cloud computing provides a computing platform for the users to meet theirdemands in an efficient, cost-effective way. Virtualization technologies areused in the clouds to aid the efficient usage of hardware. Virtual machines(VMs) are utilized to satisfy the user needs and are placed on physicalmachines (PMs) of the cloud for effective usage of hardware resources andelectricity in the cloud. Optimizing the number of PMs used helps in cuttingdown the power consumption by a substantial amount. In this paper, we present an optimal technique to map virtual machines tophysical machines (nodes) such that the number of required nodes is minimized.We provide two approaches based on linear programming and quadratic programmingtechniques that significantly improve over the existing theoretical bounds andefficiently solve the problem of virtual machine (VM) placement in datacenters.
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