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Data association is a crucial component for any multiple object tracking (MOT) method that follows the tracking-by-detection paradigm. To generate complete trajectories such methods employ a data association process to establish assignments between detections and existing targets during each timestep. Recent data association approaches try to solve either a multi-dimensional linear assignment task or a network flow minimization problem or tackle it via multiple hypotheses tracking. However, during inference an optimization step that computes optimal assignments is required for every sequence frame inducing additional complexity to any given solution. To this end, in the context of this work we introduce Transformer-based Assignment Decision Network (TADN) that tackles data association without the need of any explicit optimization during inference. In particular, TADN can directly infer assignment pairs between detections and active targets in a single forward pass of the network. We have integrated TADN in a rather simple MOT framework, designed a novel training strategy for efficient end-to-end training and demonstrated the high potential of our approach for online visual tracking-by-detection MOT on several popular benchmarks, i.e. MOT17, MOT20 and UA-DETRAC. Our proposed approach demonstrates strong performance in most evaluation metrics despite its simple nature as a tracker lacking significant auxiliary components such as occlusion handling or re-identification. The implementation of our method is publicly available at https://github.com/psaltaath/tadn-mot.