Cloud infrastructure is increasingly used for the provision of sophisticated multimedia services, such as cloud gaming or Desktop as a Service, with stringent Quality of Service demands. Serving these service demands results in the need to cost-efficiently select and capacitate data centers. In the work at hand, we introduce the corresponding Cloud Data Center Capacitation Problem and propose two optimization approaches. Through a quantitative evaluation, we demonstrate that an exact solution approach is only practically applicable to small problem instances, whereas a heuristic based on Linear Program relaxation achieves significant reductions in computation time of about 80% while retaining a favorable solution quality, with cost increases of approximately 5% or less.