Construction of balanced sports schedules using partitions into subleagues

Abstract

Direct constructions for balanced tournaments known so far solve problems with $2n$ teams if $2n\mathrm{mod}3\ne 1$ or $2n=2^p,p\ge 3$ or $n$ is odd. Our construction uses an arbitrary partition of the league into subleagues. It solves more than half of the missing cases and provides structured solutions for some known cases.

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 $2n$ teams, denoted by BTD($n$), is defined by conditions (1)–(4).

• There are $2n$ teams and $n$ different stadiums.

• Every team plays one game against every other team.

• Each team plays exactly one game each day of the tournament.

• The stadiums where a team plays are distributed as equally as possible, i.e., each team plays twice in each stadium except one.

It has been shown in [8] that BTD($n$) exists for all $2n>4$. Nevertheless, obtaining concrete solutions is another challenge. Several promising methods have been developed using tabu search [1], [5], constraint programming [11], [12] and integer programming [9]. Although these methods achieved great success in recent years, algorithmic approaches (for instance based on integer programming formulation and therefore on implicit enumeration procedures) suffer from a prohibitive computational time as the size of the problem increases (starting at $2n=16$ for basic methods and $2n=40$ for more involved ones). An alternative to this approach is to provide some direct constructions; [8] suggests solutions if $n$ is odd, [7] if $2n\mathrm{mod}3=0$ or 2 (this case is also solved in [6] by a linear time algorithm) and [3] if $2n$ is a power of 2. Notice that [3] gives more precisely solutions for a variant of BTD, denoted by BTD*, where each team plays exactly twice in each stadium. This is possible when each team plays once against every other team except one against which it plays twice; this is called a duplicated game. Here we have the additional constraint that duplicated games, involving the same teams, must not be played in the same stadium. However, it turns out that the given construction provides a schedule where all the duplicated games are played in the same day and therefore it is sufficient to remove that day in order to obtain a solution for BTD. As far as we know, the above cases include all direct constructions for BTD given in the literature so far. To summarize, the only case for which no direct construction is known is $2n\mathrm{mod}12=4$ and $2n\ne 2^p$ for some integer $p$.

In this note, we give a direct construction for BTD($n$) for specific values of $n$. In the construction method, we will arbitrarily partition the league of $2n$ 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($n$). 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($n$), 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 $2n=2^p$ with subleagues of size $2^i$ where $i$ may vary between 3 and $p-1$. Our method, although not recursive, is another construction using subleagues that may contain several even numbers of teams and not only a number $2^i$. 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 $2n$. Last but not least, it gives very structured solutions using subleagues; even though another solution is already known for some values of $2n$, 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.