We present a novel approach for multi-object tracking which considers object detection and spacetime trajectory estimation as a coupled optimization problem. It is formulated in a hypothesis selection framework and builds upon a state-of-the-art pedestrian detector. At each time instant, it searches for the globally optimal set of spacetime trajectories which provides the best explanation for the current image and for all evidence collected so far, while satisfying the constraints that no two objects may occupy the same physical space, nor explain the same image pixels at any point in time. Successful trajectory hypotheses are fed back to guide object detection in future frames. The optimization procedure is kept efficient through incremental computation and conservative hypothesis pruning. The resulting approach can initialize automatically and track a large and varying number of persons over long periods and through complex scenes with clutter, occlusions, and large-scale background changes. Also, the global optimization framework allows our system to recover from mismatches and temporarily lost tracks. We demonstrate the feasibility of the proposed approach on several challenging video sequences.