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Advanced video analytic systems, including scene classification and object detection, have seen widespread success in various domains such as smart cities and autonomous transportation. With an ever-growing number of powerful client devices, there is incentive to move these heavy video analytics workloads from the cloud to mobile devices to achieve low latency and real-time processing and to preserve user privacy. However, most video analytic systems are heavyweight and are trained offline with some pre-defined latency or accuracy requirements. This makes them unable to adapt at runtime in the face of three types of dynamism -- the input video characteristics change, the amount of compute resources available on the node changes due to co-located applications, and the user's latency-accuracy requirements change. In this paper we introduce ApproxDet, an adaptive video object detection framework for mobile devices to meet accuracy-latency requirements in the face of changing content and resource contention scenarios. To achieve this, we introduce a multi-branch object detection kernel (layered on Faster R-CNN), which incorporates a data-driven modeling approach on the performance metrics, and a latency SLA-driven scheduler to pick the best execution branch at runtime. We couple this kernel with approximable video object tracking algorithms to create an end-to-end video object detection system. We evaluate ApproxDet on a large benchmark video dataset and compare quantitatively to AdaScale and YOLOv3. We find that ApproxDet is able to adapt to a wide variety of contention and content characteristics and outshines all baselines, e.g., it achieves 52% lower latency and 11.1% higher accuracy over YOLOv3.