Lecture 20:
Sports Scheduling
© J. Christopher Beck 2005
1
Outline

ACC Basketball Scheduling



HAPs
Algorithm Flow Chart
Single Round Robin Scheduling



HAPs again
Alg 10.2.2
Example 10.2.3
© J. Christopher Beck 2005
2
ACC Scheduling

Atlantic Coast Conference
Basketball


9 teams: Clem, Duke, FSU, GT, UMD, NC,
NCSt, UVA, Wake
Double Round Robin
2 slots/week:
weekday &
weekend
Home and Away
 Total # of games to be played?
 What is the maximum # of games per slot?
 Beck
And,
3
© J. Christopher
2005 therefore the # of slots?

Constraints & Preferences

No team should play more than two
Home or two Away games consecutively



A Bye is considered an Away game
No team should play more than two
consecutive weekends Away or at Home
Each team must have at least 2 Home
or 1 Home, 1 Bye in the first 5 weeks
© J. Christopher Beck 2005
4
More Constraints &
Preferences


No team can Away for both slots in the
final week
Final weekend is usually reserved for
“rival” pairings


Duke-UNC, Clem-GT, NCSt-Wake, UMDUVA
Duke-UNC must appear in slots 9 and 18
Even with only 9 teams this is a hard problem.
Try
to decompose the solving into sub-problems.
© J. Christopher Beck
2005
5
Mirroring

Since it is a double RR, we can halve
the problem size by finding a single RR
and “mirroring” the second half

Perfect mirroring not always possible
Team 1
3
-4
2
-3
4
-2
Team 2
-4
3
-1
4
-3
1
Team 3
-1
-2
4
1
2
-4
Team 4
2
1
-3
-2
-1
3
© J. Christopher Beck 2005
6
Home Away Patterns (HAPs)

Each team has a pattern of Home &
Away games:


First (Step 1) find of a set of HAPs


HAHAAHHAAH …, AAHHAHHA …, etc.
Independent of the teams – just find
strings of Hs, As, (and maybe Bs)
Then (Step 2) match patterns to games
and finally (Step 3) assign the teams
© J. Christopher Beck 2005
7
Of Course it is More
Complicated in the Real World
38 patterns
of length 18
Find
feasible
patterns
17 pattern
sets
Find
pattern
sets
Step 1
826 timetables
17 schedules
Assign
games
Assign
teams to
patterns
Step 2
Step 3
Choose
final
schedule
Figure 10.3
© J. Christopher Beck 2005
8
Something a Bit Easier

Complete the single RR timetable

Don’t worry about Home/Away games
slot
1
2
Team a
b
f
3
5
c
Team b
a
f
Team c
d
e
Team d
c
e
Team e
f
d
c
Team f
e
a
b
© J. Christopher Beck 2005
4
a
Does this
remind you
of anything?
9
Home & Away

Now take the full time table and add
Home/Away games


Minimize breaks
Break: two
consecutive Home
or two
consecutive Away
games
© J. Christopher Beck 2005
slot
1
2
3
Team a
b
f
Team b
a
f
Team c
d
e
Team d
c
e
Team e
f
d
c
Team f
e
a
b
4
5
c
a
10
Single Round Robin
Tournament



Assume n teams and that n is even
Every team plays every other team
It is possible to construct a schedule
with n-1 slots each with n/2 games
© J. Christopher Beck 2005
11
IP for Simple Single RR

Pure IP model
xijt = 1 iff team i plays at home against
team j in slot t

n 1
n
  (x
t 1
i 1
ijt
 x jit )  1
j  1,..., n
Each team plays each other team exactly once
n 1
 (x
t 1
ijt
 x jit )  1 i  j
Each team plays exactly once in each slot
© J. Christopher Beck 2005
12
CP for Simple Single RR





xit = team that team i plays in slot t
xit є {1,…,n}
all-different
xit ≠ i
slot
1
2
3 4
xit = j  xjt = i Team a
e
all-different(xi) Team b
5
Team c
Team d
Team e
© J. Christopher Beck 2005
Team f
b
13
Simple RR Model Is
Too Simple



No optimization function
No balancing of Away/Home games
This motivates the introduction of HAPs
and the definition of breaks

Recall: a break is two consecutive games
that are both Home or both Away
© J. Christopher Beck 2005
14
What if n is Odd?


One team gets a Bye in every slot
HAPs get more complex



String of Hs, As, & Bs
Breaks need to be redefined
Can’t achieve an n-1 slot schedule

What is the minimum length schedule?
© J. Christopher Beck 2005
15
Alg 10.2.2

Step 1: Find a collection of n HAPs
Step 2: Assign a game to each entry in
the pattern set
Step 3: Assign teams to patterns

Why do we need (at least) n HAPs?


© J. Christopher Beck 2005
16
Alg 10.2.2

Step 1: Find a collection of n HAPs
Step 2: Assign a game to each entry in
the pattern set
Step 3: Assign teams to patterns

Create a 5 team single round robin




Minimize breaks (at which step?)
Now create a double RR schedule
© J. Christopher Beck 2005
17