An Application of the Traveling Tournament Problem: The Argentine Volleyball League

Since the scheduling of sports leagues involves many kinds of constraints and the minimization of costs or travel distances, sports scheduling has become a very active field, providing both interesting and challenging problems to the combinatorial optimization community. In this work we describe the automatic scheduling of the regular phase of the Argentine first division volleyball league and further research on related topics. The regular phase involves 12 teams located across the country, which play against each other following a mirrored double round-robin pattern (each team plays twice against each other, once at home and once away). In this league, the teams are grouped into 6 pairs of teams, in such a way that each pair is composed by two teams located close to each other. The matches are held on Thursdays and Saturdays, and these two matches within the same week are called a weekend. The regular phase spans 12 weekends. In each weekend, the two teams from a pair play against the two teams from another pair (two matches take place on Thursday and the two remaining matches take place on Saturday). During the weekends 1 and 7 the two teams from each pair play against themselves. Furthermore, no team can play more than two home or away weekends in a row, and every team must play at least one home weekend between weekends 2 and 3. It is important to note that, due to the involved travel distances, if a team must play two consecutive away weekends, then it does not return to its home city between these weekends, but performs a two-weekend tour instead. Finally, there are some additional constraints regarding the availability of stadiums. Our main goal was to come up with a schedule for the regular phase satisfying all the constraints and minimizing the sum of the total travel distances. This problem is a particular variation of the so-called traveling tournament problem (Easton, 2001), which has shown to be a very hard optimization problem, most of its instances still being open. On one hand, we present a natural integer programming model for this problem, which allowed to successfully generate a schedule for the Argentine first division volleyball league, that was used in the 2007/2008 Season. This result is in line with previous attempts on the traveling tournament problem based on integer programming techniques, which were able to solve to optimality instances with at most 6 teams in reasonable running times (Easton, 2003).