Virtualization is one of the core technologies used in cloud computing to provide services on demand for end users over the Internet. Most current research allocates virtual machines o physical machines based on CPU utilization. However, for many applications that require communication between services running on different servers, communication costs influence the overall performance. Therefore, this study focuses on the optimal allocation of virtual machines across multiple geographically dispersed data centers, with the objective of minimizing communication costs. The original problem can be constructed as a quadratic assignment problem that is a classical NP-hard combinatorial optimization problem. This study adopts an efficient deterministic optimization approach to reformulate the original problem as a mixed-integer linear program that may be solved to obtain a globally optimal solution. Since the required bandwidth matrix and communication cost matrix are symmetric, the mathematical model of virtual machine placement can be simplified. Several numerical examples drawn from the literature are solved to demonstrate the computational efficiency of the proposed method for determining the optimal virtual machine allocation in cloud computing.