In large-scale cloud systems, unplanned service interruptions and outages may cause severe degradation of service availability. Such incidents can occur in a bursty manner, which will deteriorate user satisfaction. Identifying incidents rapidly and accurately is critical to the operation and maintenance of a cloud system. In industrial practice, incidents are typically detected through analyzing the issue reports, which are generated over time by monitoring cloud services. Identifying incidents in a large number of issue reports is quite challenging. An issue report is typically multi-dimensional: it has many categorical attributes. It is difficult to identify a specific attribute combination that indicates an incident. Existing methods generally rely on pruning-based search, which is time-consuming given high-dimensional data, thus not practical to incident detection in large-scale cloud systems. In this paper, we propose MID (Multi-dimensional Incident Detection), a novel framework for identifying incidents from large-amount, multi-dimensional issue reports effectively and efficiently. Key to the MID design is encoding the problem into a combinatorial optimization problem. Then a specific-tailored meta-heuristic search method is designed, which can rapidly identify attribute combinations that indicate incidents. We evaluate MID with extensive experiments using both synthetic data and real-world data collected from a large-scale production cloud system. The experimental results show that MID significantly outperforms the current state-of-the-art methods in terms of effectiveness and efficiency. Additionally, MID has been successfully applied to Microsoft's cloud systems and helped greatly reduce manual maintenance effort.