Construction of balanced sports schedules using partitions into subleagues
Introduction
We consider one of the most extensively studied sports scheduling problems: balanced tournament design (BTD) on multiple venues (also known as Prob026 from CSPLib). A balanced tournament design for teams, denoted by BTD(), is defined by conditions (1)–(4).
- (1)
There are teams and different stadiums.
- (2)
Every team plays one game against every other team.
- (3)
Each team plays exactly one game each day of the tournament.
- (4)
The stadiums where a team plays are distributed as equally as possible, i.e., each team plays twice in each stadium except one.
In this note, we give a direct construction for BTD() for specific values of . In the construction method, we will arbitrarily partition the league of teams into several subleagues of equal size (i.e., number of teams). This will allow us to derive a procedure exploiting this structure. Notice that this partition is simply a tool of the construction which allows us to have a procedure which produces a structured solution to BTD(). In some cases however a partition into subleagues may be given from the beginning (based for instance on geographical requirements: each subleague may consist of teams which are located close to each other; see for example [2]). A well-known example of this is Major League Baseball in the United States (see [4]). Then, some requirements on the internal (respectively external) games (i.e., games played between teams of the same subleague (respectively, of different subleagues)) may have to be considered; for instance having in any day only internal games or only external games (see [2], [3]). Our procedure, while providing solutions to the basic problem BTD(), allows us to take also into account requirements of the above type when a partition into subleagues is given beforehand.
To our knowledge, the method described in [3] is so far the only direct construction based on a partition into subleagues: the authors give a recursive method for with subleagues of size where may vary between 3 and . Our method, although not recursive, is another construction using subleagues that may contain several even numbers of teams and not only a number . Moreover, the solutions obtained have the property that the games played each day are either all internal or all external. We provide solutions for more than half of the cases not covered by the known constructions. Another advantage of such a direct construction is that solutions can be constructed directly for arbitrarily large values of . Last but not least, it gives very structured solutions using subleagues; even though another solution is already known for some values of , having new types of schedules may be useful in order to satisfy various additional requirements, which may arise in different contexts. It is therefore important to have a wide sample of solutions, among which to choose the best one.
Section snippets
Balanced tournament design using subleagues
BTD() can be alternatively formulated as an arrangement of the distinct pairs of teams in the set of teams into a array in such a way that every element of occurs exactly once in each row (day) and at most twice in each column (stadium).
Let us recall that a Graeco-Latin square of order is an array where the entries contain all distinct ordered pairs of elements from two sets of size in such a way that each element appears once in each row and once in each column.
Results and discussion
Clearly, Table 7 can be extended to infinity while leaving some holes, i.e. values of for which no direct construction is known so far. Programming approaches (integer programming or constraint programming) leave no such holes but they are practically limited to relatively small values of (see for instance [5], [11]).
As can be observed in Table 7, our construction gives BTD() for all odd and for all values of such that a Graeco-Latin square of order is known and BTD() can be
Acknowledgement
The research of Tınaz Ekim is supported by Grant 200020-113405 of the Swiss National Science Foundation.
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