The ever-increasing use of artificial intelligence in autonomous systems has significantly contributed to advance the research on multi-object tracking, adopted in several real-time applications (e.g., autonomous driving, surveillance drones, robotics) to localize and follow the trajectory of multiple objects moving in front of a camera. Current tracking algorithms can be divided into two main categories: some approaches introduce complex heuristics and re-identification models to improve the tracking accuracy and reduce the number of identification switches, without particular attention to the timing performance, whereas other approaches are aimed at reducing response times by removing the re-identification phase, thus penalizing the tracking accuracy. This work proposes a new approach to multi-class object tracking that allows achieving smaller and more predictable execution times, without penalizing the tracking performance. The idea is to reduce the problem of matching predictions with detections into smaller sub-problems by splitting the Hungarian matrix by class and invoking the second re-identification stage only when strictly necessary for a smaller number of elements. The proposed solution was evaluated in complex urban scenarios with several objects of different types (as cars, trucks, bikes, and pedestrians), showing the effectiveness of the multi-class approach with respect to state of the art trackers.