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© J. Christopher Beck 2005 2
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 3
ACC Scheduling
Atlantic Coast ConferenceBasketball 9 teams: Clem, Duke, FSU, GT, UMD, NC,
NCSt, UVA, Wake Double Round Robin
Home and Away Total # of games to be played? What is the maximum # of games per
slot? And, therefore the # of slots?
2 slots/week:weekday &
weekend
© J. Christopher Beck 2005 4
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 5
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,
UMD-UVA Duke-UNC must appear in slots 9 and
18Even with only 9 teams this is a hard problem.
Try to decompose the solving into sub-problems.
© J. Christopher Beck 2005 6
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
Team 2
-4 3 -1
Team 3
-1 -2 4
Team 4
2 1 -3
-3 4 -2
4 -3 1
1 2 -4
-2 -1 3
© J. Christopher Beck 2005 7
Home Away Patterns (HAPs)
Each team has a pattern of Home & Away games: HAHAAHHAAH …, AAHHAHHA …, etc.
First (Step 1) find of a set of HAPs 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 8
Of Course it is More Complicated in the Real World
Findfeasiblepatterns
Findpattern
sets
Assigngames
Assignteams topatterns
Choosefinal
schedule
38 patternsof length 18
17 patternsets
826 timetables 17 schedules
Step 1 Step 2 Step 3
Figure 10.3
© J. Christopher Beck 2005 9
Something a Bit Easier
Complete the single RR timetable Don’t worry about Home/Away games
slot 1 2 3 4 5
Team a b f c
Team b a f
Team c d e a
Team d c e
Team e f d c
Team f e a b
Does thisremind youof anything?
© J. Christopher Beck 2005 10
Home & Away
Now take the full time table and add Home/Away games
slot 1 2 3 4 5
Team a b f c
Team b a f
Team c d e a
Team d c e
Team e f d c
Team f e a b
Minimize breaks Break: two
consecutive Home or two consecutive Away games
© J. Christopher Beck 2005 11
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 12
IP for Simple Single RR
njxxn
ijitijt
n
t
,...,11)(1
1
1
jixxn
tjitijt
1
1
1)(
Each team plays each other team exactly once
Each team plays exactly once in each slot
Pure IP model xijt = 1 iff team i plays at home
against team j in slot t
© J. Christopher Beck 2005 13
CP for Simple Single RR
xit = team that team i plays in slot t
xit є {1,…,n} xit ≠ i xit = j xjt = i all-different(xi)
slot 1 2 3 4 5
Team a
Team b
Team c
Team d
Team e
Team f
all-different
e
b
© J. Christopher Beck 2005 14
Simple RR Model IsToo 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 15
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 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
Why do we need (at least) n HAPs?