Breakout Group Allocation Schedules and the Social Golfer Problem with Adjacent Group Sizes
Abstract
:1. Introduction
2. Good, Balanced Allocation Schedules
- 1.
- there are at most 2 group sizes,
- 2.
- group sizes differ by at most 1,
- 3.
- every round has the same distribution of group sizes,
- 4.
- if all groups have a size of at least 3
- 5.
- If and there are two group sizes, then all groups have sizes between 4 and 6.
3. The Social Golfer Problem and a Variation
32 golfers play golf once a week, and always in groups of 4. For how many weeks can they play such that no two players play together more than once in the same group?
The Social Golfer Problem with Adjacent Group Sizes
4. Finding Solutions to the SGA Using Results from Combinatorial Design Theory
4.1. Preliminary Definitions And Examples
4.2. Constructing Solutions for Sga
- 1.
- If a exists then k divides v and divides .
- 2.
- 3.
- If there is a solution to where and where, for every round, there are blocks of size k and blocks of size , then r is at most , the largest integer less than or equal to .
- 4.
- If and there is no solution to with [50].
- 5.
- There is no solution to for (a consequence of [51]).
- ,
- there are at most 2 block sizes,
- block sizes differ by at most 1,
- every round has the same distribution of block sizes, i.e., either all rounds have blocks of the same size (k) or they have blocks of size k and blocks of size , where and are the same for all rounds
- If and there are two block sizes then all blocks have size between 4 and 6.
- r is the largest for which a solution has been found.
5. Good, Balanced Allocation Schedules
5.1. At Most Eleven Participants
5.2. Between Twelve and Nineteen Participants
5.3. Between Twenty and Fifty Participants
5.4. More than Fifty Participants
6. Obtaining Solutions from Our Website
7. Discussion
Author Contributions
Funding
Conflicts of Interest
References
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Notation | Description |
---|---|
use a | |
use a with p points removed from a single block in the final round | |
use a with p points removed from a single block in the final round and the final round of blocks removed | |
use a | |
use the blocks of an | |
use the blocks of an with p points removed from a single group | |
use the best available solution to the social golfer problem for v players in groups of size k | |
use the best available solution to the social golfer problem for v players in groups of size k, with p points removed, no pair of which appear in a block of the | |
as above, but remove the points from final block and remove the final round of blocks | |
use an existing design D (a Kirkman system or resolvable transversal design, say), with p points added, each to a single block in each round, where these blocks don’t intersect, for as many rounds as possible |
v | K | Solution | r | ||
---|---|---|---|---|---|
12 | 3 | 4 | |||
13 | 5 | 4 | |||
14 | 5 | 4 | |||
15 | 3 | 7 | 7 | ||
5 | 5 | ||||
16 | 4 | 5 | 5 | ||
6 | 5 | ||||
17 | 6 | 5 | |||
18 | 3 | 8 | 8 | ||
6 | 5 | ||||
19 | 6 | 5 | |||
8 | 6 |
v | K | Solution | r | ||
---|---|---|---|---|---|
20 | 4 | 5 | |||
21 | 3 | 10 | 10 | ||
6 | 5 | ||||
22 | 6 | 5 | |||
23 | 6 | 5 | |||
24 | 3 | 11 | 11 | ||
4 | 7 | 7 | |||
6 | 6 | ||||
25 | 5 | 6 | 6 | ||
7 | 5 | ||||
26 | 7 | 5 | |||
27 | 3 | 13 | 13 | ||
7 | 5 | ||||
28 | 4 | 9 | 9 | ||
7 | 6 | ||||
29 | 7 | 6 | |||
8 | 7 | ||||
30 | 3 | 14 | 14 | ||
5 | 7 | 6 | |||
8 | 7 | ||||
31 | 8 | 7 | |||
7 | 3 | ||||
32 | 4 | 10 | 10 | ||
8 | 7 | ||||
7 | 3 | ||||
33 | 3 | 16 | 16 | ||
8 | 7 | ||||
10 | 8 | ||||
7 | 3 | ||||
34 | 8 | 7 | |||
10 | 8 | ||||
7 | 3 | ||||
35 | 5 | 8 | 7 | ||
9 | 8 | ||||
7 | 3 | ||||
36 | 3 | 17 | 17 | ||
4 | 11 | 8 | |||
6 | 3 | ||||
9 | 8 | ||||
8 | 7 | ||||
37 | 9 | 8 | |||
11 | 9 | ||||
8 | 7 | ||||
38 | 9 | 8 | |||
11 | 9 | ||||
8 | 7 | ||||
39 | 3 | 19 | 19 | ||
9 | 8 | ||||
11 | 9 | ||||
8 | 7 | ||||
40 | 4 | 13 | 13 | ||
5 | 9 | 8 | |||
11 | 9 | ||||
8 | 7 |
v | K | Solution | r | ||
---|---|---|---|---|---|
41 | 11 | 9 | |||
12 | 9 | ||||
8 | 7 | ||||
9 | 8 | ||||
42 | 3 | 20 | 20 | ||
6 | 8 | 7 | |||
11 | 9 | ||||
12 | 6 | ||||
9 | 8 | ||||
43 | 11 | 9 | |||
12 | 5 | ||||
9 | 8 | ||||
44 | 4 | 14 | 11 | ||
11 | 9 | ||||
12 | 7 | ||||
9 | 8 | ||||
45 | 3 | 22 | 22 | ||
5 | 11 | 9 | |||
12 | 7 | ||||
14 | 11 | ||||
9 | 8 | ||||
46 | 12 | 7 | |||
13 | 11 | ||||
9 | 8 | ||||
10 | 9 | ||||
47 | 12 | 7 | |||
13 | 11 | ||||
9 | 8 | ||||
10 | 9 | ||||
48 | 3 | 23 | 23 | ||
4 | 15 | 12 | |||
6 | 9 | 8 | |||
12 | 7 | ||||
13 | 11 | ||||
10 | 9 | ||||
49 | 7 | 8 | 8 | ||
12 | 7 | ||||
13 | 11 | ||||
15 | 12 | ||||
10 | 9 | ||||
50 | 5 | 12 | 7 | ||
13 | 11 | ||||
15 | 12 | ||||
10 | 9 |
v | K | Solution | r | ||
---|---|---|---|---|---|
6 | 2 | 5 | 5 | ||
7 | 4 | 3 | |||
8 | 2 | 7 | 7 | ||
4 | 4 | ||||
9 | 3 | 4 | 4 | ||
6 | 4 | ||||
10 | 2 | 5 | 9 | 9 | |
5 | 4 | ||||
11 | 5 | 4 | |||
7 | 5 |
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Miller, A.; Barr, M.; Kavanagh, W.; Valkov, I.; Purchase, H.C. Breakout Group Allocation Schedules and the Social Golfer Problem with Adjacent Group Sizes. Symmetry 2021, 13, 13. https://doi.org/10.3390/sym13010013
Miller A, Barr M, Kavanagh W, Valkov I, Purchase HC. Breakout Group Allocation Schedules and the Social Golfer Problem with Adjacent Group Sizes. Symmetry. 2021; 13(1):13. https://doi.org/10.3390/sym13010013
Chicago/Turabian StyleMiller, Alice, Matthew Barr, William Kavanagh, Ivaylo Valkov, and Helen C. Purchase. 2021. "Breakout Group Allocation Schedules and the Social Golfer Problem with Adjacent Group Sizes" Symmetry 13, no. 1: 13. https://doi.org/10.3390/sym13010013
APA StyleMiller, A., Barr, M., Kavanagh, W., Valkov, I., & Purchase, H. C. (2021). Breakout Group Allocation Schedules and the Social Golfer Problem with Adjacent Group Sizes. Symmetry, 13(1), 13. https://doi.org/10.3390/sym13010013