Lecture 21:
Sports Scheduling 1
© J. Christopher Beck 2008
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Outline
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ACC Basketball Scheduling


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HAPs
Algorithm Flow Chart
Single Round Robin Scheduling
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

HAPs again
Alg 10.2.2
Example 10.2.3
© J. Christopher Beck 2008
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Readings
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
P Ch 10.6,10.2
Questions

10.1, 10.2, 10.4,
10.5, 10.6, 10.8
© J. Christopher Beck 2008
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ACC Scheduling
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Atlantic Coast Conference
Basketball
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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,
4
© J. Christopher
2008 therefore the # of slots?

Constraints & Preferences
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No team should play more than two
Home or two Away games consecutively
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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 2008
5
More Constraints &
Preferences
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
No team can be Away for both slots in
the final week
Final weekend is usually reserved for
“rival” pairings
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
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
2008
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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 2008
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Home Away Patterns (HAPs)
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Each team has a pattern of Home &
Away games:
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First (Step 1) find of a set of HAPs
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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 2008
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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 2008
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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 2008
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a
Does this
remind you
of anything?
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Home & Away
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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 2008
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
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Single Round Robin
Tournament
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
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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 2008
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IP for Simple Single RR
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Pure IP model
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n
 (x
i 1
ijt
xijt = 1 iff team i plays at home against
team j in slot t
 x jit )  1
j  1,..., n; t  1,, n  1
Each team plays exactly once in each slot
n 1
 (x
t 1
ijt
 x jit )  1 i  j
Each team plays each other team exactly once
© J. Christopher Beck 2008
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CP for Simple Single RR



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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 2008
Team f
b
14
Simple RR Model Is
Too Simple
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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 2008
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What if n is Odd?
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One team gets a Bye in every slot
HAPs get more complex
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String of Hs, As, & Bs
Breaks need to be redefined
© J. Christopher Beck 2008
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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
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Why do we need (at least) n HAPs?
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
© J. Christopher Beck 2008
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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
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Create a 5 team single round robin
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
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Minimize breaks (at which step?)
Now create a double RR schedule
© J. Christopher Beck 2008
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Next Week
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We start to read some papers
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These are real papers, published in the
research literature.
You should not expect to completely
understand them in the first reading.
You should read them (at least once)
before lecture and (at least once) after.
1 next week, 2 week after, 1 more later
© J. Christopher Beck 2008
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