||The proliferation of innovative mobile services such as augmented reality, networked gaming, and autonomous driving has spurred a growing need for low-latency access to computing resources that cannot be met solely by existing centralized cloud systems. Mobile Edge Computing (MEC) is expected to be an effective solution to meet the demand for low- latency services by enabling the execution of computing tasks at the network edge, in proximity to the end-users. While a number of recent studies have addressed the problem of determining the execution of service tasks and the routing of user requests to corresponding edge servers, the focus has primarily been on the efficient utilization of computing resources, neglecting the fact that non-trivial amounts of data need to be pre-stored to enable service execution, and that many emerging services exhibit asymmetric bandwidth requirements. To fill this gap, we study the joint optimization of service placement and request routing in dense MEC networks with multidimensional constraints. We show that this problem generalizes several well-known placement and routing problems and propose an algorithm that achieves close- to-optimal performance using a randomized rounding technique. Evaluation results demonstrate that our approach can effectively utilize available storage, computation, and communication resources to maximize the number of requests served by low- latency edge cloud servers.