We propose a novel approach to automatically detecting and tracking cell populations in time-lapse images. Unlike earlier ones that rely on linking a predetermined and potentially under-complete set of detections, we generate an overcomplete set of competing detection hypotheses. We then perform detection and tracking simultaneously by solving an integer program to find an optimal and consistent subset. This eliminates the need for heuristics to handle missed detections due to occlusions and complex morphology. We demonstrate the effectiveness of our approach on a range of challenging image sequences consisting of clumped cells and show that it outperforms state-of-the-art techniques.