Relax-fix-optimize heuristics for time-relaxed sports timetabling

Time-relaxed sports timetables utilize (many) more time slots than there are games per team, and therefore offer the flexibility to take into account player and venue availability. However, time-relaxed tournaments also have the drawback that the difference in games played per team and the rest period between teams' consecutive games can vary considerably. In addition, organisers may want to avoid consecutive home and away games (i.e., breaks). To construct fair timetables, we propose relax-and-fix (R&F) and fix-and-optimize (F&O) heuristics that make use of team- and time-based variable partitioning schemes. While the team-based R&F constructs a timetable by gradually taking into account the integrality constraints related to all home games of a subset of teams, the time-based R&F maintains a rolling horizon of time intervals in which the integrality constraints of all games scheduled within the time interval are activated. The F&O heuristics use the same variable partitioning schemes, but they never relax the integrality constraints and allow to recover from mistakes by making a small number of changes with respect to the variables optimized in previous iterations. For numerous real-life instances, our heuristics generate high-quality timetables using only a fraction of the computational resources used by monolithic integer programming solvers