The research of this work is motivated by the Timetable Synchronization Problem in public transport. The problem is formulated as a multi-criteria Quadratic Semi-Assignment Problem (QSAP). This type of problem is known to be NP-hard. Therefore, the real life problem instances are solved with metaheuristics. To evaluate the quality of the solutions, lower bounds are generated by using the Reformulation Linearization Technique (RLT). This work contains new polyhedral results for the QSAP and analyzes the characteristics of the RLT solutions. A graph structure that causes untight solutions is presented and its minimality is proven. Exploiting these results, the natural stepwise structure of the RLT can be softened up to generate new algorithms for fast lower bound computations. The competitiveness of the approach is demonstrated by means of the real life instances.