# Breakout Group Allocation Schedules and the Social Golfer Problem with Adjacent Group Sizes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Good, Balanced Allocation Schedules

**Definition**

**1.**

- 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 $v\ge 12$ all groups have a size of at least 3
- 5.
- If $v\ge 20$ and there are two group sizes, then all groups have sizes between 4 and 6.

## 3. The Social Golfer Problem and a Variation

`sci.op-research`:

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?

**Lemma**

**1.**

#### The Social Golfer Problem with Adjacent Group Sizes

**Definition**

**2.**

**Lemma**

**2.**

## 4. Finding Solutions to the SGA Using Results from Combinatorial Design Theory

#### 4.1. Preliminary Definitions And Examples

**Example**

**1.**

**Example**

**2.**

#### 4.2. Constructing Solutions for Sga

**Lemma**

**3.**

- 1.
- If a $KS(v,k)$ exists then k divides v and $k-1$ divides $v-1$.
- 2.
- 3.
- If there is a solution to $SGA(v,K,r)$ where $K=\{k,k+1\}$ and where, for every round, there are ${m}_{1}>0$ blocks of size k and ${m}_{2}\ge 0$ blocks of size $k+1$, then r is at most $R(v,k,{m}_{1},{m}_{2}))$, the largest integer less than or equal to $v(v-1)/\left(k\right(m1(k-1)+m2(k+1))$.
- 4.
- If $k=4$ and $v=20$ there is no solution to $SGP(20,4,r)$ with $r=R(20,4,5,0)$ [50].
- 5.
- There is no solution to $SGP(36,6,r)$ for $r>3$ (a consequence of [51]).

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

- $12\le v\le 50$,
- 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 ${m}_{1}$ blocks of size k and ${m}_{2}$ blocks of size $k+1$, where ${m}_{1}$ and ${m}_{2}$ are the same for all rounds
- If $v\ge 20$ 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 |
---|---|

$KS(v,k)$ | use a $KS(v,k)$ |

$KS(v,k)-p$ | use a $KS(v,k)$ with p points removed from a single block in the final round |

$KS(v,k)-p,B$ | use a $KS(v,k)$ with p points removed from a single block in the final round and the final round of blocks removed |

$NKTS\left(v\right)$ | use a $NKTS\left(v\right)$ |

$RTD(k,n)$ | use the blocks of an $RTD(k,n)$ |

$RTD(k,n)-p$ | use the blocks of an $RTD(k,n)$ with p points removed from a single group |

$SG(v,k)$ | use the best available solution to the social golfer problem for v players in groups of size k |

$SG(v,k)-p$ | 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 $SG(v,k)$ |

$SG(v,k)-p,B$ | as above, but remove the points from final block and remove the final round of blocks |

$D+p$ | 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 |

**Table 2.**Solutions to $SGA(v,K,r)$ where $r\ge 3$, $12\le v\le 19$, and $K=\left\{3\right\},\left\{4\right\}$ or $\{3,4\}$.

v | K | $\mathit{m}1,\mathit{m}2$ | $\mathit{MAX}$ | Solution | r |
---|---|---|---|---|---|

12 | 3 | $4,0$ | ${4}^{*}$ | $KS(16,4)-4,B$ | 4 |

13 | $3,4$ | $3,1$ | 5 | $KS(16,4)-3,B$ | 4 |

14 | $3,4$ | $2,2$ | 5 | $KS(16,4)-2,B$ | 4 |

15 | 3 | $5,0$ | 7 | $KS(15,3)$ | 7 |

$3,4$ | $1,3$ | 5 | $KS(16,4)-1$ | 5 | |

16 | 4 | $4,0$ | 5 | $KS(16,4)$ | 5 |

$3,4$ | $4,1$ | 6 | $RTD(4,5)-4$ | 5 | |

17 | $3,4$ | $3,2$ | 6 | $RTD(4,5)-3$ | 5 |

18 | 3 | $6,0$ | 8 | $NKTS(18,3)$ | 8 |

$3,4$ | $2,3$ | 6 | $RTD(4,5)-2$ | 5 | |

19 | $3,4$ | $1,4$ | 6 | $RTD(4,5)-1$ | 5 |

$5,1$ | 8 | $NKTS\left(18\right)+1$ | 6 |

**Table 3.**Solutions to $SGA(v,K,r)$ where $r\ge 3$, $20\le v\le 40$, and $K=\left\{3\right\},\left\{4\right\},\left\{5\right\},\{4,5\}$ or $\{5,6\}$.

v | K | $\mathit{m}1,\mathit{m}2$ | $\mathit{MAX}$ | Solution | r |
---|---|---|---|---|---|

20 | 4 | $5,0$ | ${5}^{*}$ | $RTD(4,5)$ | 5 |

21 | 3 | $7,0$ | 10 | $KS(21,3)$ | 10 |

$4,5$ | $4,1$ | 6 | $KS(25,5)-4,B$ | 5 | |

22 | $4,5$ | $3,2$ | 6 | $KS(25,5)-3,B$ | 5 |

23 | $4,5$ | $2,3$ | 6 | $KS(25,5)-2,B$ | 5 |

24 | 3 | $8,0$ | 11 | $NKTS(24,3)$ | 11 |

4 | $6,0$ | 7 | $KS(24,4)$ | 7 | |

$4,5$ | $1,4$ | 6 | $KS(25,5)-1$ | 6 | |

25 | 5 | $5,0$ | 6 | $KS(25,5)$ | 6 |

$4,5$ | $5,1$ | 7 | $SG(30,5)-5,B$ | 5 | |

26 | $4,5$ | $4,2$ | 7 | $SG(30,5)-4,B$ | 5 |

27 | 3 | $9,0$ | 13 | $KS(27,3)$ | 13 |

$4,5$ | $3,3$ | 7 | $SG(30,5)-3,B$ | 5 | |

28 | 4 | $7,0$ | 9 | $KS(28,4)$ | 9 |

$4,5$ | $2,4$ | 7 | $SG(30,5)-2$ | 6 | |

29 | $4,5$ | $1,5$ | 7 | $SG(30,5)-1$ | 6 |

$6,1$ | 8 | $RTD(5,7)-6$ | 7 | ||

30 | 3 | $10,0$ | 14 | $NKTS(30,3)$ | 14 |

5 | $6,0$ | 7 | $SG(30,5)$ | 6 | |

$4,5$ | $5,2$ | 8 | $RTD(5,7)-5$ | 7 | |

31 | $4,5$ | $4,3$ | 8 | $RTD(5,7)-4$ | 7 |

$5,6$ | $5,1$ | 7 | $SG(36,6)-5$ | 3 | |

32 | 4 | $8,0$ | 10 | $SG(32,4)$ | 10 |

$4,5$ | $3,4$ | 8 | $RTD(5,7)-3$ | 7 | |

$5,6$ | $4,2$ | 7 | $SG(36,6)-4$ | 3 | |

33 | 3 | $11,0$ | 16 | $KS(33,3)$ | 16 |

$4,5$ | $2,5$ | 8 | $RTD(5,7)-2$ | 7 | |

$7,1$ | 10 | $RTD(5,8)-7$ | 8 | ||

$5,6$ | $3,3$ | 7 | $SG(36,6)-3$ | 3 | |

34 | $4,5$ | $1,6$ | 8 | $RTD(5,7)-1$ | 7 |

$6,2$ | 10 | $RTD(5,8)-6$ | 8 | ||

$5,6$ | $2,4$ | 7 | $SG(36,6)-2$ | 3 | |

35 | 5 | $7,0$ | 8 | $RTD(5,7)$ | 7 |

$4,5$ | $5,3$ | 9 | $RTD(5,8)-5$ | 8 | |

$5,6$ | $1,5$ | 7 | $SG(36,6)-1$ | 3 | |

36 | 3 | $12,0$ | 17 | $NKTS(36,3)$ | 17 |

4 | $9,0$ | 11 | $SG(36,4)$ | 8 | |

6 | $6,0$ | ${3}^{*}$ | $SG(36,6)$ | 3 | |

$4,5$ | $4,4$ | 9 | $RTD(5,8)-4$ | 8 | |

$5,6$ | $6,1$ | 8 | $RTD(6,7)-6$ | 7 | |

37 | $4,5$ | $3,5$ | 9 | $RTD(5,8)-3$ | 8 |

$8,1$ | 11 | $RTD(5,9)-8$ | 9 | ||

$5,6$ | $5,2$ | 8 | $RTD(6,7)-5$ | 7 | |

38 | $4,5$ | $2,6$ | 9 | $RTD(5,8)-2$ | 8 |

$7,2$ | 11 | $RTD(5,9)-7$ | 9 | ||

$5,6$ | $4,3$ | 8 | $RTD(6,7)-4$ | 7 | |

39 | 3 | $13,0$ | 19 | $KS(39,3)$ | 19 |

$4,5$ | $1,7$ | 9 | $RTD(5,8)-1$ | 8 | |

$6,3$ | 11 | $RTD(5,9)-6$ | 9 | ||

$5,6$ | $3,4$ | 8 | $RTD(6,7)-3$ | 7 | |

40 | 4 | $10,0$ | 13 | $KS(40,4)$ | 13 |

5 | $8,0$ | 9 | $RTD(5,8)$ | 8 | |

$4,5$ | $5,4$ | 11 | $RTD(5,9)-5$ | 9 | |

$5,6$ | $2,5$ | 8 | $RTD(6,7)-2$ | 7 |

**Table 4.**Solutions to $SGA(v,K,r)$ where $r\ge 3$, $41\le v\le 50$ and $K=\left\{3\right\},\left\{4\right\},\left\{5\right\},\left\{6\right\},\left\{7\right\},\{4,5\}$ or $\{5,6\}$.

v | K | $\mathit{m}1,\mathit{m}2$ | $\mathit{MAX}$ | Solution | r |
---|---|---|---|---|---|

41 | $4,5$ | $4,5$ | 11 | $RTD(5,9)-4$ | 9 |

$9,1$ | 12 | $KS(40,4)+1$ | 9 | ||

$5,6$ | $1,6$ | 8 | $RTD(6,7)-1$ | 7 | |

$7,1$ | 9 | $RTD(6,8)-7$ | 8 | ||

42 | 3 | $14,0$ | 20 | $NKTS(42,3)$ | 20 |

6 | $7,0$ | 8 | $RTD(6,7)$ | 7 | |

$4,5$ | $3,6$ | 11 | $RTD(5,9)-3$ | 9 | |

$8,2$ | 12 | $KS(40,4)+2$ | 6 | ||

$5,6$ | $6,2$ | 9 | $RTD(6,8)-6$ | 8 | |

43 | $4,5$ | $2,7$ | 11 | $RTD(5,9)-2$ | 9 |

$7,3$ | 12 | $KS(40,4)+3$ | 5 | ||

$5,6$ | $5,3$ | 9 | $RTD(6,8)-5$ | 8 | |

44 | 4 | $11,0$ | 14 | $RTD(5,11)-11$ | 11 |

$4,5$ | $1,8$ | 11 | $RTD(5,9)-1$ | 9 | |

$6,4$ | 12 | $SG(50,5)-6$ | 7 | ||

$5,6$ | $4,4$ | 9 | $RTD(6,8)-4$ | 8 | |

45 | 3 | $15,0$ | 22 | $KS(45,3)$ | 22 |

5 | $9,0$ | 11 | $RTD(5,9)$ | 9 | |

$4,5$ | $5,5$ | 12 | $SG(50,5)-5$ | 7 | |

$10,1$ | 14 | $RTD(5,11)-10$ | 11 | ||

$5,6$ | $3,5$ | 9 | $RTD(6,8)-3$ | 8 | |

46 | $4,5$ | $4,6$ | 12 | $SG(50,5)-4$ | 7 |

$9,2$ | 13 | $RTD(5,11)-9$ | 11 | ||

$5,6$ | $2,6$ | 9 | $RTD(6,8)-2$ | 8 | |

$8,1$ | 10 | $RTD(6,9)-8$ | 9 | ||

47 | $4,5$ | $3,7$ | 12 | $SG(50,5)-3$ | 7 |

$8,3$ | 13 | $RTD(5,11)-8$ | 11 | ||

$5,6$ | $1,7$ | 9 | $RTD(6,8)-1$ | 8 | |

$7,2$ | 10 | $RTD(6,9)-7$ | 9 | ||

48 | 3 | $16,0$ | 23 | $NKTS(48,3)$ | 23 |

4 | $12,0$ | 15 | $RTD(4,12)$ | 12 | |

6 | $8,0$ | 9 | $RTD(6,8)$ | 8 | |

$4,5$ | $2,8$ | 12 | $SG(50,5)-2$ | 7 | |

$7,4$ | 13 | $RTD(5,11)-7$ | 11 | ||

$5,6$ | $6,3$ | 10 | $RTD(6,9)-6$ | 9 | |

49 | 7 | $7,0$ | 8 | $KS(49,7)$ | 8 |

$4,5$ | $1,9$ | 12 | $SG(50,5)-1$ | 7 | |

$6,5$ | 13 | $RTD(5,11)-6$ | 11 | ||

$11,1$ | 15 | $RTD(5,12)-11$ | 12 | ||

$5,6$ | $5,4$ | 10 | $RTD(6,9)-5$ | 9 | |

50 | 5 | $10,0$ | 12 | $SG(50,5)$ | 7 |

$4,5$ | $5,6$ | 13 | $RTD(5,11)-5$ | 11 | |

$10,2$ | 15 | $RTD(5,12)-10$ | 12 | ||

$5,6$ | $4,5$ | 10 | $RTD(6,9)-4$ | 9 |

**Table 5.**Solutions to $SGA(v,K,r)$ where $r\ge 3$, $6\le v<12$ and $K=\left\{2\right\},\left\{3\right\}$ or $\{2,3\}$.

v | K | $\mathit{m}1,\mathit{m}2$ | $\mathit{MAX}$ | Solution | r |
---|---|---|---|---|---|

6 | 2 | $3,0$ | 5 | $SG(6,2)$ | 5 |

7 | $2,3$ | $2,1$ | 4 | $KS(9,3)-2,B$ | 3 |

8 | 2 | $4,0$ | 7 | $SG(8,2)$ | 7 |

$2,3$ | $1,2$ | 4 | $KS(9,3)-1$ | 4 | |

9 | 3 | $3,0$ | 4 | $KS(9,3)$ | 4 |

$2,3$ | $3,1$ | 6 | $SG(8,2)+1$ | 4 | |

10 | 2 | 5 | 9 | $SG(10,2)$ | 9 |

$2,3$ | $2,2$ | 5 | $SG(8,2)+2$ | 4 | |

11 | $2,3$ | $1,3$ | 5 | $KS(16,4)-5,B$ | 4 |

$4,1$ | 7 | $SG(10,2)+1$ | 5 |

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Miller, 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