1. © J. Christopher Beck 20051 Lecture 20: Sports Scheduling

2. © 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

3. © J. Christopher Beck 2005 3 ACC Scheduling Atlantic Coast Conference Basketball 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

4. © 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

5. © 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 18 Even with only 9 teams this is a hard problem. Try to decompose the solving into sub-problems.

6. © 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 13-42 Team 2-43 Team 3-24 Team 421-3 4-2 4-31 12-4 -23

7. © 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

8. © J. Christopher Beck 2005 8 Of Course it is More Complicated in the Real World Find feasible patterns Find pattern sets Assign games Assign teams to patterns Choose final schedule 38 patterns of length 18 17 pattern sets 826 timetables17 schedules Step 1 Step 2Step 3 Figure 10.3

9. © J. Christopher Beck 2005 9 Something a Bit Easier Complete the single RR timetable Don’t worry about Home/Away games slot12345 Team abfc Team baf Team cdea Team dce Team efdc Team feab Does this remind you of anything?

10. © J. Christopher Beck 2005 10 Home & Away Now take the full time table and add Home/Away games slot12345 Team abfc Team baf Team cdea Team dce Team efdc Team feab Minimize breaks Break: two consecutive Home or two consecutive Away games

11. © 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

12. © J. Christopher Beck 2005 12 IP for Simple Single RR Each team plays each other team exactly once Each team plays exactly once in each slot Pure IP model x ijt = 1 iff team i plays at home against team j in slot t

13. © J. Christopher Beck 2005 13 CP for Simple Single RR x it = team that team i plays in slot t x it є {1,…,n} x it ≠ i x it = j  x jt = i all-different(x i ) slot12345 Team a Team b Team c Team d Team e Team f all-different e b

14. © J. Christopher Beck 2005 14 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

15. © 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?

16. © 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?

17. © J. Christopher Beck 2005 17 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