Matthew Lai
CS 537 Scheduling
Sports Scheduling
Written by Kelly Easton, George Nemhauser, Michael Trick
Paper Summary
Introduction
Sports scheduling is the process of scheduling games between multiple teams into what the
paper calls a tournament. The paper focuses on Round robin tournaments, specifically the single round
robin and double round robin tournaments. The single round robin tournament consists of games
scheduled such that each team plays each other team exactly once. Double round robin tournaments
consists of games in which each team plays each other team twice, usually once at each team’s home
venue. Other tournament types mentioned and expanded on in the paper is the balanced tournament
design problem consisting of games that are hosted within neutral grounds with common facilities to
host all games, the bipartite tournament problem which states the problem of scheduling games
between the members of two groups such that each member of one group plays all the members in the
other group, and the traveling tournament problem which states a problem in which the home stands
and road trips are considered as well as the total distance by each team.
There are some useful terms that should be introduced prior to the paper proper, some of
which has cropped up in the previous paragraph. A break is the number of times that a team has
consecutive home or away games. A tournament is a series of games in which teams play each other
team. A schedule is a mapping of games into slots such that each team plays at most once in each slot. A
mirrored or partially mirrored schedule occurs in double round robin games in which a team’s schedule
can simply be mirrored so that each team plays the other teams twice without having to recalculate the
schedule; the teams will play each team once at each venue. A compact schedule is a schedule that
includes the minimum possible number of slots. A pattern is a vector of home, away, or bye designations
for a single team over the slots of a schedule. A complementary pattern is such that one pattern consists
of slots that are exactly complimentary to the respective slots in the other pattern. A pattern set is a
collection of patters, one for each team. A tour is the schedule for a single team. A trip or road trip is a
series of consecutive away games. A home stand is the opposite, it is a series of consecutive home
games.
In the first part of this paper we will look at defining the problems that were introduced in the
chapter. We will discuss the specific solutions for each problem and the specific details of each. We will
then discuss some of the more general solutions to sports scheduling including graph algorithms, integer
programming, constraints programming, and some metaheuristic algorithms.
Round Robin Tournaments
Single Round Robin Tournament
The single round robin tournament is a tournament in which each team plays each other team
exactly once. It can be formally described with the following definition:
Input: A set of n teams T = { 1, 2, …, n }
Output: A mapping of the games in the set G = {gij : i, j T, i < j}
To slots in the set S = {sk, k = 1, …, n-1 if n is even and k = 1, …, n if n is odd}
This SSRT problem can be described as a block design problem which is described as (v, k, ). The
block design algorithm attempts to create subsets of size k using the v elements of the set where each
element can only appear in  subsets. The SSRT problem, therefore, can be described as (n, 2, 1) where n
is the number of teams participating in the tournament, 2 is the size of each subset, and 1 is the number
of subsets that a team can appear simultaneously in. In other words, each of the n teams is placed into a
subset of size 2 and in at most 1 subset. That subset is then a
scheduled game for that slot. This block design problem can
be solved with what is known as the circle method. This
method is described as follows: Each teams is labeled in order
Figure 1 An example of the Circle method
𝑛
𝑛
as , 1, 2, …, n-1. On day i, play team i vs. , (i-1) vs (i+1) …, (i − 2 − 1) vs (i + 2 − 1). Each integer is
reduced by (mod n – 1) to keep the team number within the interval [1, n -1]. The  label is then replaced
with team n. Figure 1 is an example with n = 8. Team n or 8 is paired against team i, team 1 on day 1 and
team 2 on day 2. This continues for every team in the set. This ensures that each team plays each other
team only once within the tournament.
Another solution that can be applied to many round robin tournaments, including the SSRTP, is
the Latin Squares problem. A Latin square is an n x n array that is
filled with n different symbols each of which occur only once in each
row, and only once in each column. By creating a Latin square where
each row and column of the array is a representation of a team, one
can create a schedule for a tournament. In each cell is the slot in
which the two teams would play each other. Figure 2 is such an
example of scheduling teams into a Latin square. For day one, the
Figure 2 Example of a Latin Square
Schedule
game schedules is simply the teams that correspond to the cells with a value of one, in this case the
upward diagonal from the bottom left of the square. There has been a lot of research done in the field of
Latin squares and their construction and much of that research can be adopted and applied into the sports
scheduling field.
Pattern Set Problems
An alternative way to look at scheduling problems is with pattern sets. In our previous problem
statement and solutions, schedules were constructed by assigning team matchups to schedule slots. The
reverse process has also been suggested to be an effective alternative to this train of thought. In this case,
a pattern set is generated for each team in which they have available slots in which they can play games.
This might be a pattern of once every few days or once every week. Then compatible teams are paired
with these pattern sets to form schedules. It is then imperative to find a good algorithm to generate
pattern sets that will match among different teams. Pattern sets that are generated in this matter will be
either feasible patterns, patterns where the set of teams will successfully generate a round robin
tournament, or infeasible patterns, where the various teams’ patterns cannot be matched to form a
tournament. The whole set of constraints that must be met in order to generate feasible patterns is not
currently known, however some basic constraints is useful as baseline constraints. For example, each pair
of patterns must differ by at least one slot. This means that any pattern can only appear at most once in
a pattern set. For double round robin tournaments, the conditions are stronger. For each pattern pairs
generated for any two teams, there must be at least one slot in which one team is at home and the other
team is away, and another slot where the team’s play venue is reversed. This is to ensure that teams will
eventually match back up with the other team to play each team twice, once in each venue. Another
condition for round robin, is that every slot in the pattern set must include an equal number of home and
away games. This is obvious because there must be one team at home and one team playing away
Balanced Tournament Designs
A balanced tournament is, as mentioned previously in this paper, a tournament where all the
games are played entirely within a neutral set of venues. This means that each game in the tournament
has an additional constraint, the slot availability of the venues themselves. The formal definition of the
balanced tournament design problem is as follows. The input is a set of n teams T = { 1, …, n } and a number
of facilities F. The problem then is to map the games in order to get an output of games in set
G = {gij : i, j ∈ T, i < j } to the slots available at each facility described by the set S = {sfk, f = 1, …, F, k = 1, …,
n – 1 if n is even and k = 1, …, n if n is odd } such that no more than one game involving team i is assigned
to a particular slot and the difference between the number of appearances of team i at two separate
facilities is no more than 1. In other words, a set of games, each with a pair of teams, must be mapped to
the available time slots at a set of facilities. No team can play in more than one game in any one slot and
the number of appearances for each team must be spread across all facilities.
One proposed solution to this problem is the BTDP-NO algorithm. This “bracelet” algorithm works
for 2m+1 odd number of teams. To perform the BTDP-NO algorithm, each team is arranged from 1 through
2m + 1 into an elongated pentagon or “bracelet”. Each facility is
associated with each row containing two teams. For each slot k= 1, …,
2m+1 give the team at the top of the pentagon the bye. For each row
with two teams, i, j associated with facility f, assign gij to skf. Each team is Figure 3 An example of the BTDP-NO
algorithm
then shifted one position in clockwise direction for the next day. Figure
3 is an example of m=2 with 5 different teams. In this tournament, the first set of games will be between
teams 2 and 5, 3 and 4, each playing within each the m facilities, facility 1 and facility 2 respectively for
this first set. Team 1 will receive the bye game in which they do not play. On the next day each team is
rotated clockwise such that team 2 receives the bye and team 1 plays team 3, team 4 plays team 5, each
playing in facility 1 and facility 2 respectively.
Bipartite Tournament
The bipartite tournament is a tournament in which there are two sets of teams or players. Each
individual in each set must play every individual in the other set. The formal definition of the bipartite
tournament is as follows. The input for this tournament is two teams with n players (or teams)
T1 = { x1, ..., xn } and T2 = { y1, …, yn }. The output for this tournament type is a mapping of games in the
set G = { gij : i ∈ T1, j ∈ T2 }, to the slots in the set S = {sk, k = 1, …, n } such that exactly one game including
t is mapped to any given slot for all t ∈ T1  T2. As was stated, this problem is not necessarily limited to
players, but also to teams, encompassing the concept of leagues or conferences of teams.
General Solution Methods for Sports Scheduling
Graph Algorithms
The sports scheduling problem can be solved as a graph problem. The sports schedule would be
represented on a graph, with 2m teams, as a graph K2m with 2m colors. The edge [i, j] would represent a
game between team i and team j. The SRRT problem can be
presented as a 1-factorization of K2m such that each vertex (or
team) is not connected by any two edges of the same value (or
color). In this way, by taking the 1-factor of a certain value (or
color), you receive a perfect matching of pairs such that no vertex
(or team) is connected to more than one other vertex. This 1factorized graph should be oriented (made directed) such that the Figure 4 An example color graph
direction indicates the home/away orientation of the game. In figure 4, we have a 1-factorization graph
with 5 colors. Each node has, at most, one edge of each color. In this
graph, a team might be represented as a number, with six teams present
within this graph. To find the schedule of a slot is the simple matter of
taking the 1-factor of ‘red’ to get a pairing between the teams. Because
the graph is a 1-factorization graph, no node will be scheduled to more
Figure 5 1-factor of figure 4
than one game in any time slot. Figure 5 gives an example of the 1-factor
of red for the graph given in figure 4. To schedule venues in addition to tournament game scheduling, the
graph simply becomes an oriented graph where each edge becomes a directed edge, as shown in figure 5.
For double round robin tournament, an extra edge with a symmetric orientation to the existing edge is
added between any two nodes. In this way, the teams will play each other twice, once in each venue.
In general, sports scheduling algorithms, such as the aforementioned graph solution, will be
adapted to attempt to solve several different sports scheduling problems. One of these problems is the
Minimum Breaks problem, an attempt to minimize the total number of breaks the team within a
tournament receives. In SRRTP, with an even number of teams, it is possible, albeit with only two feasible
patterns, to have a tournament with no breaks. That is no team will have a home stand or trip greater
than 1. Therefore with an even number of teams in SRRTP, the tournament will most likely be scheduled
with multiple breaks for each team in order to satisfy other conditions. With an odd number of teams,
there is the introduce of a bye for a single team in every slot, thus increasing the number of feasible
patterns available, allowing for many more patterns with no breaks. Another problem that many general
algorithms attempt to solve is the Geographical Location problem. An important constraint concerning
geographical location is not necessarily ensuring that teams minimize travel, but to ensure that the
scheduling patterns generated do no result in consecutive games in which the two teams with similar
geographical locale play each other consecutively in each venue. This is to increase attendance by
introducing different matchups in a shorter period of time, as attendees are more likely to attend to see
another team play rather than the same matchup. A third sports scheduling problem that will be described
in this section is the Divisions problem. Sports leagues often organize themselves into divisions. Teams
within a division are often organized by geography in which division teams are relatively close to one
another when compared to intra-division matchups. Thus many leagues, such as the MLB, add the
scheduling constraint that teams play inter-division games on the weekday and intra-division games on
the weekend. De Werra, who wrote many papers on the subject of sports scheduling, goes more into
detail with oriented factorizations of complete graphs to solve this problem.
Integer Programming
Integer programming is a mathematical optimization or feasibility program where some or all
variables are restricted to be integers. Integer programming is an NP-hard problem that was adapted and
developed by Nemhauser and Trick - authors of the paper on which this paper is summarizing - for sports
scheduling. An integer programming solution for SRRTP is easy to formulate. Let binary variables xijk = 1 if
i plays j in slot k and xijk = 0 otherwise for i < j ∈ {1, …, n } and k ∈ { 1,… n – 1} for even n. In Basic SRRTP the
IP problem should satisfy the two following constraints.
∑ 𝑥𝑖𝑗𝑘 = 1,
∀𝑖 ∈ 1, … , 𝑛,
𝑘 ∈ 1, … , 𝑛 − 1
𝑗:𝑗>𝑖
∑ 𝑥𝑖𝑗𝑘 = 1∀𝑖,
𝑘
𝑥𝑖𝑗𝑘 ,
∀𝑗 ∈ 1, … , 𝑛,
𝑖<𝑗
𝑏𝑖𝑛𝑎𝑟𝑦 ∀𝑖, 𝑗 ∈ {1, … , 𝑛},
𝑖 < 𝑗,
𝑘 ∈ 1, … , 𝑛 − 1
The first constraint guarantees that every team plays exactly once in each slot. The second ensures that
each team plays every opponent exactly once.
Constraint Programming
Constraints Programming is a programming paradigm with a combinatorial approach for solving
hard optimization problems. CP for sports scheduling, as implemented by Martin Henz, performed the
steps differently from IP. It first assigned teams to patterns before matching teams in games. This
constraints programming solution for sports programming reported ran on the order of minutes as
compared to the 24 hour run time from the Integer Programming.
Traveling Tournament Problem
The traveling tournament problem is a unique tournament problem proposed by Nemhauser and
Trick that can be solved with their unique hybrid integer programming and constraints programming
solution. Its input is a set of teams T = { 1, …, n }; D: an n x n integer distance matrix with elements dij; l, u
integer parameters. The output is a double round robin tournament in which the teams in set T follow
certain constraints. The length of each home stand and road trip is between l and u inclusive. The total
distance traveled by any team is also minimized. With the maximum value of u = (n – 1) , this problem
becomes a traveling salesman problem. For a small u the team must return home often, increasing travel
distance. For u = 1, the problem becomes on of finding a feasible solution.
To solve this problem, a hybrid method using both Integer Programming and Constraints
Programming was used. These two different models are used in a parallel algorithm known as the parallel
branch and price algorithm. Each individual team tour is represented as the columns of this column
generating algorithm. The Constraints Programming is used to solve the pricing problem to determine the
solution to the minimum price, in this case, travel distance. The Integer Processor will attempt to
determine the tours for each team in the tournament that will best solve the home stand/road trip
constraint of the problem. Additionally a CP model is also used as part of a primal heuristic that is run on
a separate processor on the side that will attempt to look for a good, but not necessarily best solution by
using the solutions generated so far by the other parts of the algorithm.
Metaheuristics: Simulated Annealing
Simulated Annealing is a generic probabilistic metaheuristic for global optimization problem of
locating a good approximation to
the global optimum of a given
function in a large search space.
The algorithm starts from a
current state S. It probabilistically
Figure 6 Simulated Annealing Example
decides between moving from the current state to some neighboring S’ state. In this TTSA algorithm, the
regions containing infeasible schedules penalize the algorithm to discourage it from remaining in the area.
The figure 6 gives an example of a stage of simulated annealing. At this stage with a higher temperature,
the marker that is searching for the global maximum will move at a fast rate. As the temperature drops,
the marker will decrease its rate of jump and its location will be increasingly narrowed to the points that
the algorithm has determined is the most statistically likely location of the global maximum.
Conclusion
The scheduling field of sports scheduling contains many different possible problems that begs the
proposal of different solutions. The round robin tournament is only one type of scheduling problem that
is present in the world of sports. In reality there remain many different types of tournaments that each
must be solved using other means. Many scheduling problems might be solved by a specific solution.
Others might be solvable with a more general solution, such as the Latin squares scheduling solution. And
there also exists yet more solutions that base their problem solving on the various ways to construct
constraints that govern how a schedule is formed. These include Integer Programming and Constraints
Programming. And another possible solution is the use of heuristic algorithms that attempt to find the
best solution without necessarily finding the most optimal solution. Overall, the field of sports scheduling,
and indeed the world of scheduling in general, is a vast and difficult world. The problems are many and
the resources to solve these problems are generally few.
Resources
Easton, Kelly, George Nemhauser, and Michael Trick. "Sports Scheduling." Handbook of Scheduling:
Algorithms, Models, and Performance Analysis. Ed. Joseph Y-T Leung. Boca Raton: Chapman & Hall,
2004. 52-1. Print.