Combinatorial Solutions to the Social Golfer Problem and the Social Golfer Problem with Adjacent Group Sizes

Abstract

The Social Golfer problem (SGP) consists of scheduling v players into rounds of equally sized groups in such a way that (1) any two players are assigned to the same group in at most one round and (2) as many rounds as possible are obtained. Combinatorial properties dictate the maximum theoretical number of rounds that may or may not be achievable. Any solution with the theoretically maximum number of rounds is called a maximal solution, and solutions with the number of rounds that is the best currently known (but not necessarily maximal) are said to be optimal. Existing techniques to find optimal solutions consist of exhaustive search methods and constructions based on combinatorial structures such as mutually orthogonal Latin squares (MOLSs) and mutually orthogonal Latin rectangles (MOLRs). In this paper, we investigate other combinatorial designs that can provide optimal solutions with at least as many rounds as those published and introduce novel constructions based on transversal designs, incomplete transversal designs, and starter blocks. We also provide optimal solutions to a related problem, where group sizes may differ by one but all rounds have the same number of groups of each size (the Social Golfer problem with adjacent group sizes (SGA)). We show how optimal solutions to this problem can be derived from optimal solutions to an instance of the SGP with either more or fewer players. An algorithm is presented to find an optimal solution in general, and solutions are provided for up to 150 players.

1. Introduction

The study of combinatorial designs [1,2] is a well-established area of mathematical research. A combinatorial design is an arrangement of a set of elements into defined substructures (blocks) in such a way that desired combinatorial properties are satisfied. They have many applications in, for example, communications, cryptography and networking [3], optical orthogonal codes [4], cancer trial design [5], genetic screening algorithms [6], and cloud computing [7].

In this paper, we are mainly interested in designs for which every pair of elements appears in at most one block together and one of the desirable properties is resolvability, i.e., where the blocks can be arranged into rounds in which every element appears once. Such designs are called resolvable designs [8].

The focus of this paper is the link between resolvable designs and another mathematical problem: the Social Golfer problem (SGP). The Social Golfer problem is to allocate v players into groups of equal size over as many rounds as possible. No two players should play together in more than one round, and the goal is to allocate as many rounds as possible. Any solution with the theoretically maximum number of rounds is called a maximal solution, and solutions with the number of rounds that is the best currently known (but not necessarily maximal) are are said to be optimal. Of course, solutions that may have once been optimal may have been superseded (and hence are no longer optimal). We note that the terms used in SGP (such as group) conflict with similar terms used in combinatorial designs. In Section 2, we restate the problem using combinatorial design terminology.

Note that maximal solutions to SGP are examples of a class of combinatorial designs known as resolvable maximum packings [9,10].

A survey of approaches to find optimal SGP solutions was given in [11]. These include design theoretic approaches based on sets of mutually orthogonal Latin squares (MOLSs). The link between SGP and sets of MOLSs was identified in [12]. Specifically, for certain instances, if there are enough MOLSs, maximal SGP solutions can be obtained from structures known as resolvable balanced incomplete block designs. Other constructions, based on less-restrictive structures called orthogonal Latin rectangles (MOLRs), were introduced in [13]. A ten-round (maximal) solution for 32 players in groups of size 4 design (see Section 2) originally constructed by Shen [14] and independently solved by Colbourn in 1999 [15] was later shown to be equivalent to the rounds of blocks of a structure known as a resolvable group divisible design [16].

Most of the research into SGP focuses on the development of efficient search algorithms to find as many rounds as possible for SGP. The goal in this case is to compare the performance of proposed algorithms with previous approaches in terms of the number of rounds obtained and how quickly, rather than in the solutions themselves.

One such approach is to use heuristics. For example, in [17], greedy search is enhanced by introducing a heuristic based on the intuitive concept of freedom among players. Similarly, in [18], greedy search is further improved by removing cliques from the search space during search. Another approach, presented in [19] and improved on in [20], uses large neighbourhood search in a declarative problem-solving setting to find solutions to hard problems (such as SGP). Boolean satisfiability (SAT) has also been used. A SAT encoding of SGP was originally proposed in [21] and enhanced in [22,23].

Due to the highly symmetric nature of SGP, it is used as a benchmark for the development of efficient constraints-based methods that exploit these symmetries [24]. We give some examples where symmetry breaking is used to enhance search and SGP is used to demonstrate the effectiveness of the approach.

Harvey and Winterer [25] developed efficient constraints-based techniques using symmetry breaking to find maximal sets of MOLRs and hence find (then) optimal SGP solutions in some instances. In [26], constraints were added during search to avoid the exploration of symmetric paths. In [27], a collection of techniques were proposed specifically for finding optimal solutions to SGP. In [28], value precedence on integer and set sequences was used to remove some symmetry from the related constraints satisfaction problem. Search space splitting and data-level parallelism were exploited in [29] to enhance search.

Combinatorial search, however well tuned, is an inefficient way to find optimal solutions to SGP and is, therefore, limited to a relatively small number of players (typically 20). However, the best-possible solutions found by these approaches allow us to determine whether our own solutions are indeed optimal. A list of results using search methods or common design theoretic techniques is given at [30]. However, this has not been updated since 2002.

In [31], we considered the problem of finding optimal SGP solutions for up to 50 players. We also considered the case where there are two group sizes that differ by one and, in each round, the number of groups of each size is constant. We called this the Social Golfer problem with adjacent group sizes (SGA). This is very helpful in situations where the total number of participants does not have any suitable divisors.

We provided an initial list of (then) optimal solutions for up to 50 participants and group sizes, either a divisor of the class size or a pair of consecutive values ($(4, 5)$ or $(5, 6)$). In most cases, we found our solutions by adapting resolvable designs, in particular, resolvable balanced incomplete block designs and a particular type of resolvable group divisible designs, namely, resolvable transversal designs. In some cases, we used (or adapted) existing published optimal SGP solutions. We also made all our solutions available on our BoRAT web application, the link to which can be found at [32]. We continue to add solutions as they are generated.

The motivation for this paper came from our observations in [31]. We realised that it was important not only to find solutions but also to investigate the combinatorial structures that can be exploited to find them. In this paper, we re-introduce the SGP and the SGA and present examples of the combinatorial designs that we use to find optimal (and in many cases maximal) solutions to them.

Our first contribution is a summary of the resolvable combinatorial structures that are used in our constructions (Section 3).

Our second contribution is to describe constructions that provide optimal solutions to SGP and SGA. These include new constructions based on (incomplete) transversal designs (Section 4), previously published constructions using mutually orthogonal Latin squares (MOLSs) and mutually orthogonal Latin rectangles (MOLRs) (Section 5), and a new construction using starter blocks (also Section 5).

Our third contribution is an algorithm to determine which method to use to find an optimal solution when groups are of equal size (Section 6). When groups have size 3, the algorithm returns a maximal solution (one that cannot be improved) for any number of participants. When groups have size 4, it returns a maximal solution in all but a handful of cases (where the solution has one less round than the theoretical maximum). Otherwise, the algorithm suggests an optimal solution. For group sizes of at least 5, our algorithm is only guaranteed to provide an optimal solution for up to 150 players. For more players, it is likely that additional corner cases would need to be considered.

We discuss how to construct optimal solutions for unequal group sizes (4 and 5 or 5 and 6) by adding participants to smaller optimal SGP solutions or removing them from larger ones.

Our final contribution (Section 8) is to improve solutions given in [31] and extend our results for SGP and SGA (with group sizes 4 and 5 or 5 and 6) for up to 150 participants. We highlight some difficult cases, presenting the method used to find the current optimal solution.

Complete tables containing SGP and SGA solutions for up to 150 participants are provided in the Supplementary Materials.

2. Social Golfer Problem and the Social Golfer Problem with Adjacent Block Sizes

2.1. The Social Golfer Problem

The Social Golfer problem (which we refer to as SGP) [11,30,33] originated from the following question posed in 1998 to 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?

In fact, the answer to this problem is ten weeks. We will show in Section 6.2 that the solution can be obtained using a combinatorial structure known as a Resolvable Group Divisible Design. The design for this case was originally constructed by Shen in [14] and independently solved by Colbourn in 1999 [15]. In 2004, Aguado recognised that the design would solve the problem and published a solution [16].

The problem can be generalised as follows (see [29]):

The SGP consists of scheduling $n=g\ast s$ players into g groups of s players for w weeks so that any two players are assigned to the same group at most once in w weeks.

The SGP has been applied in a variety of contexts. For example, it has been used in group teaching scenarios. In [31], it was used to allocate students to breakout rooms in Zoom meetings. In [34,35] it is used to assign students to groups for an interactive virtual reality game and for a group assignment, respectively. SGP has been used as a method to assign participants to deliberation groups in a study of voting strategies [36]. It is also used in scheduling for sporting competitions such as volleyball [37,38], football [39], and Mahjong [40].

Any instance of the SGP exhibits four types of symmetry arising from permuting the weeks, groups within weeks, players within groups, and the players within the overall set of players.

Due to its highly symmetric nature, SGP has been adopted by the combinatorial search community as a benchmark for testing advanced search techniques that exploit symmetry. These techniques allow solutions to combinatorial problems to be found faster by breaking, removing, and discarding symmetry during search. We have summarised some of these appproaches in Section 1.

In this paper, rather than relying on optimised search, we use existing combinatorial designs to construct solutions.

We will adopt a notation that is slightly different from the one used in the SGP literature. As we will be using combinatorial designs to generate rounds, we refer to blocks rather than groups (the term group already plays an important role in designs, as we will see in Section 3). We also refer to rounds rather than weeks and points instead of players. We will continue to use the terms players and groups until we redefine the problem in Section 2.2 (Definition 2).

We focus on optimal solutions: i.e., solutions to problems for which no solution with more rounds is known. We hence distinguish between optimal and maximal solutions:

Definition 1.

A solution to SGP$(n, k)$ with r rounds is said to be an optimal solution if no solution with more than r rounds is known. A solution is maximal if no solution with more than r rounds is possible.

If a solution is maximal, it must be optimal. As a player can be in a group with every other player at most once, and every group that contains a given player contains $s-1$ other players, a player can be in at most $r=\lfloor (n-1)/(s-1)\rfloor$ rounds. Hence, if a solution has r rounds, it is clearly maximal.

Consider the example shown in Figure 1. There are 28 players (labelled 0 to 27), and each round consists of seven groups of size 4. There are nine rounds, and the solution is clearly maximal.

Now consider the example shown in Figure 2. There are 24 players, and each round consists of six groups of size 4. There are seven rounds, and the solution is again maximal.

Note that, in the example shown in Figure 1, all pairs of players appear in the groups. However, in Figure 2, this is not the case (for example, player 0 is not in a group with either player 8 or player 16). As we will see in Section 5, this is because the example in Figure 1 is from a Resolvable Balanced Incomplete Block Design (RBIBD), and the example in Figure 2 is from (the blocks of) a Resolvable Group Divisible Design (RGDD).

2.2. The Social Golfer Problem with Adjacent Block Sizes

In [31], we introduced a problem closely related to SGP, which we call the Social Golfer problem with adjacent block sizes (SGA). This problem arises when allocating players to groups when there is no suitable group size that exactly divides the number of players, in particular, when the number of players is prime. In this case, we do not insist that all groups have the same size. Obviously, if all group sizes are possible, there would be far too many solutions. A reasonable first restriction is to have two group sizes that differ by as little as possible, i.e., they differ by 1. We denote the set of group sizes by K.

As we will show in Section 7, solutions can be obtained in these cases by examining a solution to SGP with more players and removing certain sets of players (or, in some cases, by adding players to solutions with fewer players).

For example, if we remove players 0 and 8 from the example in Figure 2, we obtain an allocation where the groups have sizes 3 and 4 (as we already knew that 0 and 8 were not in an existing group, it comes as no surprise that no group loses more than one player). To obtain a legitimate allocation (with players in set $\{x:0 \le x \le 21\}$), we renumber the players in the obvious way. See Figure 3.

As the problem initially arose in the context of assigning students to small groups, for which group sizes between 4 and 6 are considered optimal [41], in [31], we further restricted ourselves to the cases $K = \{4, 5\}$ and $K=\{5, 6\}$. We will continue to focus on these cases in this paper, although the techniques we introduce can easily be extended to further block sizes.

The link between the Social Golfer problem and combinatorial designs, specifically mutually orthogonal Latin squares [1] [Table VI.16.54], Kirkman systems [1] [II.2.8] [42,43], and Resolvable Group Divisible Designs [1] [IV.5.3], is well known [11,25,27]. Some optimal solutions to the Social Golfer problem can be found in [30,33].

Because we will derive our solutions to SGP and SGA from suitable combinatorial designs, to avoid confusion, we will henceforth adopt Definitions 2 and 3 for SGP and SGA in terms of points and blocks (rather than players and groups):

Definition 2.

The SGP$(v, k, r)$ consists of arranging $v = k\ast n$ points into r rounds of n non-intersecting blocks of size k in such a way that two points are assigned to the same block at most once.

Definition 3.

The SGA$(v, n_1, k_1, n_2, k_2, r)$ consists of arranging $v = n_1\ast k_1 + n_2\ast k_2$ points into r rounds of $n_1 + n_2$ non-intersecting blocks in such a way that

1. 

$k_2 = k_1 + 1$;

2. 

each round contains $n_1$ blocks of size $k_1$ and $n_2$ blocks of size $k_2$;

3. 

two points are assigned to the same block at most once.

Generally, the Social Golfer Problem refers to fixed values of v and k and finding the largest r for which SGP$(v,k,r)$ has a solution. In the context of allocation schedules (e.g., for sports competitions), a maximal solution may not be required (as often only a few rounds are required, or the maximal solution would provide too many rounds to feasibly implement). However, finding the maximal solution (or the optimal solution) will allow any number of rounds $r^{\prime }$ up to this value to be obtained by choosing only the first $r^{\prime }$ rounds:

Lemma 1.

If there is a solution for SGP$(v,k,r)$, then there is a solution for SGP$(v,k,r^{\prime })$ for any $r^{\prime }$ less than r.

Remark 1.

When discussing an instance of SGP (i.e., for a given v and k) for which we are trying to find an optimal solution, we simply refer to SGP$(v,k)$.

3. Combinatorial Structures: PBDs, BIBDs, GDDs, and URDs

In this section, we introduce the combinatorial structures that will be required for our allocations in Section 6, Section 7 and Section 8. It is not intended to be a complete review of all combinatorial structures. In order to maintain consistency, we adapt the definitions given in [1] where possible, although they appear in slightly different forms in various publications. A combinatorial design is an arrangement of a set of elements into defined substructures (blocks) in such a way that desired combinatorial properties are satisfied. They have many applications in, for example, communications, cryptography and networking [3], optical orthogonal codes [4], cancer trial design [5], genetic screening algorithms [6], and cloud computing [7]. Our starting point is pairwise balanced designs.

Definition 4.

Let K be a set of positive integers and let λ be a positive integer. A pairwise balanced design (PBD) of order v with block sizes from K is a pair $(V,B)$, where V is a finite set (the point set) of cardinality v and B is a family of subsets (blocks) of V that satisfy (1) if $b\in B$ then $\left|b\right|\in K$ and (2) every pair of distinct elements of V occurs in exactly λ blocks of B. When $\lambda =1$, we refer to such a PBD as a PBD$(v,K)$.

In this paper, we only consider PBDs for which $\lambda =1$, so in the remaining definitions, we will only include this case. The following definition is for a BIBD, which is a special type of PBD with $\left|K\right|=1$.

Definition 5.

A balanced incomplete block design (BIBD) is a PBD where all blocks have the same size. If the block size is k and the design is on v points (i.e., has order v), we refer to a k-BIBD and to a specific BIBD with parameters v and k as a BIBD$(v,k)$. If the blocks can be arranged into classes (or rounds) in which every class contains every point exactly once (i.e., parallel classes), then the BIBD is said to be resolvable and is called a resolvable balanced incomplete block design (referred to as an RBIBD, a k-RBIBD, or an RBIBD$(v,k)$).

A 3-RBIBD is known as a Kirkman triple system [42,43,44,45].

For a BIBD$(v,k)$, the parameters v and k are as defined above; in addition, $b = \left[v\right(v-1\left)\right]/\left[k\right(k-1\left)\right]$ and $r = (v-1)/(k-1)$ denote the number of blocks and the number of blocks containing any given point (the regularity), respectively. In addition, if the BIBD is resolvable, then there are r rounds, each of which contains $v/k$ blocks.

Lemma 2, stated in [46] [Theorem 1.1], gives two necessary (but not always sufficient) conditions for the existence of a BIBD$(v, k)$. The conditions are sufficient if $3 \le k \le 5$ [46] [Lemmas 5.4, 5.11 and 5.19].

Lemma 2.

If there exists a BIBD$(v, k)$, then

1. 

$v-1\equiv 0 (mod (k-1\left)\right)$;

2. 

$v(v-1)\equiv 0( mod k(k-1\left)\right)$.

Surveys of existence results for BIBDs with a block size at most 9 are given in [47] and [48] [Table 1].

We focus here on RBIBDs as they are of most use to us in the context of solutions for the Social Golfer problem.

Lemma 3.

If $3 \le k \le 5$ and an RBIBD$(v, k)$ exists, then

1. 

$k = 3$ and $v \equiv 3 (mod 6)$ [43], or

2. 

$k = 4$ and $v \equiv 4 (mod 12)$ [49], or

3. 

$k = 5$ and $v \equiv 5 (mod 20)$, except possibly when$v \in \{45, 345, 465, 645\}$ [50,51,52,53,54,55].

For an RBIBD$(v,6)$ to exist, we require $v = 30t + 6$ for some integer $t \ge 0$. Many of the constructions for RBIBDs for $6 \le k \le 9$ rely on a theorem using difference families (see Definition 6) due to Ray-Chaudhuri and Wilson [56] (also stated in [1] [Remark II.7.23]), which we state as Theorem 1. Note that, for prime power q, $G_q$ denotes the finite field of order q.

Definition 6.

Let G be a group of order v. A $(v,k,1)$ difference family is a set of k-subsets $B = \{B_1, B_2, \dots , B_t\}$ of G, where the differences $(a-b\mid a,b\in B_i;a\ne b, i = 1\dots t)$ contain each non-zero member of G exactly once.

Theorem 1.

If q is an odd prime power, and a block disjoint $(q, k, 1)$ difference family over $G_q$ exists, then so does an RBIBD$(kq, k)$.

Some results for the existence of RBIBD$(v, 6)$ are given in Theorem 2.

Theorem 2.

1. 

There is no RBIBD$(36, 6)$. This follows from [8] [Theorem 4.1.4].

2. 

An RBIBD$(156, 6)$ exists. The construction of such a design is given in [57].

3. 

If q is a prime power less than $2000$, not equal to $61$ or $121$, and $q \equiv 1 (mod 30)$, then there is a block-disjoint $(q, 6, 1)$ difference family [58] [Theorem 11], see also [1] [Table VI.16.54]. By Theorem 1, an RBIBD$(6q, 6)$ exists for these cases.

4. 

An RBIBD$(1716, 6)$ exists and a construction is given in [59] [Theorem 12.8].

5. 

If q is a prime power, $q \equiv 1 (mod 10)$, $q\not\equiv 1 (mod 50)$, $q \ne 11$, and $q < 1500$, then an RBIBD$(6q, 6)$ exists [60] [Lemma 3.3].

6. 

If $t \le 832$ is even, $t\notin \{2, 6, 8, 12\}$, and $6t + 1$ is a prime power, then there is an RBIBD$(30t + 6, 6)$ [61] [Theorem 5.1].

7. 

There exists an RBIBD$(125q + 1, 6)$ for any prime $q \equiv 7 (mod 12)$ and $q>43$ and for any prime $q \equiv 1 (mod 12)$ and $q > 37$ [62] [Theorem 18].

Note that although there are very few RBIBD$(v, 6)$ designs for $v = 30t + 6$ and t odd, for $t \le 100$, some larger such designs with t odd can be constructed by noting that if an RBIBD$(w, 6)$ and an RTD$(6, w)$ (see Definition 10) exist, then so does an RBIBD$(6w, 6)$. For example, for $v = 936$ ($t = 31$), an RBIBD$(v, 6)$ can be constructed this way.

A description of methods for the construction of RBIBD$(v, k)$ for $k\in \{7, 8, 9\}$ is given in [59]. We summarise known results for these cases in Theorem 3.

Theorem 3.

1. 

Lists of $(q, k, 1)$ block disjoint difference families for odd prime powers q and $7 \le k \le 9$ are given in [60] [Tables C.I–C.III] and [63] [Tables A.1–A.5].

2. 

A construction for a $(577, 9, 1)$ block disjoint difference family is given in [64].

3. 

If t is one of the following $25$ integers less than $100$, an $RBIBD(42t + 7,7)$ exists ([59] [Table C.1]): $0,1,8,9,17,28,33,41,49,56,57,63,64,65,70,72,73,77,80,81,88,89,91,96,97$.

4. 

Constructions for $RBIBD(56t + 8, 8)$ for all but 66 values of t are given in [65], and constructions for eight further values of t are given in [62].

5. 

If t is  not  one of the values in $\{3,$ $13,$ $20,$ $22,$ $23,$ $25,$ $26,$ $27,$ $31,$ $38,$ $43,$ $47,$ $52,$ $58,$ $59,$ $61,$ $67,$ $69,$ $76,$ $79,$ $93\}$ and $t\le 100$, then an $RBIBD(56t + 8, 8)$ exists.

6. 

If t is one of the $16$ integers less than $100$ in $\{0,$ $1,$ $7,$ $8,$ $9,$ $10,$ $37,$ $54,$ $64,$ $71,$ $72,$ $73,$ $81,$ $82,$ $90,$ $91\}$, then an $RBIBD(72t + 9, 9)$ exists ([59] [Table C.1]).

Remark 2.

The only RBIBD$(v,k)$ with $k > 9$ that is used for our allocations is RBIBD$(121, 11)$. This can be constructed using an RTD$(11, 11)$ (see Definition 10).

A PBD that is particularly useful to us is one in which a distinguished set of blocks, called groups, forms a parallel class. This is known as a group divisible design (GDD). We only consider the case where all blocks have the same size (a k-GDD) and in most cases where the groups have the same size (a uniform k-GDD). More general group divisible designs are used in the constructions for GDDs whose blocks are resolvable—namely, resolvable group divisible designs (RGDDs). We provide the necessary definition below.

Definition 7.

A group divisible design is a triple $(V,B,G)$, where V is a set of points, B and G are sets of subsets of V (called blocks and groups, respectively), (1) every pair of distinct elements of V occurs in exactly one block or one group, but not both, and (2) the groups partition the set of points V. A k-GDD of type $g_0^{u_0}, g_1^{u_1}, \dots g_{s-1}^{u_{s-1}}$ is a GDD with blocks of size k and $u_i$ groups of size $g_i$, for $0\le i\le s-1$. A uniform k-GDD of type $g^u$ is a k-GDD with u groups of size g. If the blocks of a uniform GDD can be arranged into parallel classes, then the GDD is called a resolvable group divisible design (a k-RGDD). If the blocks of a GDD can be arranged into partial parallel classes, each of which misses just the points in a single group, then the GDD is called a frame.

Since, in an RGDD, every point appears in a block with every other point that does not lie in the same group as it, a k-RGDD with groups of size g has $(v - g)/(k - 1)$ rounds. Notice that a k-RBIBD is a k-RGDD of type $k^{v/k}$ (the last round of blocks provides the groups).

Lemma 4, from [66] [Lemma 1.1], implies that if we are trying to use the blocks of an RGDD to generate rounds of blocks of equal size, we must use a uniform RGDD.

Lemma 4.

If an RGDD has blocks of the same size, then its groups are of the same size.

For our allocations, the only RGDDs with blocks of size 3 that we require (that are not a KTS) are those with groups of size 2. A 3-RGDD of type $2^{v/2}$ is known as a nearly Kirkman triple system (NKTS$\left(v\right)$) [45,67,68,69,70,71,72,73]. The existence result in Lemma 5 is from [45] [Theorem 27].

Lemma 5.

There exists an NKTS$\left(v\right)$ if and only if $v\equiv 0 (mod 6)$ and $v\notin \{6, 12\}$.

When $k = 4$ and $v \equiv 0$ or $8 (mod 12)$, we will use 4-RGDDs with group size 3 or 2, respectively, to produce maximal allocations. We, therefore, limit our discussion of 4-RGDDs to these cases.

The existence result in Lemma 6 for 4-RGDDs with groups of size 3 follows from constructions in [9,14,72,74,75,76,77,78].

Lemma 6.

If $v\equiv 0 (mod 12)$ and $u = v/3$, then a 4-RGDD of type $3^u$ exists, except when $u = 4$.

The existence result in Lemma 7 for 4-RGDDs with groups of size 2 follows from constructions in [9,14,78,79,80,81,82,83].

Lemma 7.

If $v\equiv 8 (mod 12)$ and $u = v/2$, then a 4-RGDD of type $2^u$ exists, except when $u = 4$, $u = 10$, and possibly when $u\in \{46, 70, 82, 94, 100, 118, 130, 202, 214\}$.

We use 4-RGDDs for some of our allocations in Section 6. The constructions for them are often recursive and can involve the construction of many smaller, intermediate designs.

The existence of many of the combinatorial structures we use to provide SGP solutions relies on the existence of sufficient numbers of mutually orthogonal Latin squares (MOLSs). The following definitions are from [1] [Chapter III]:

Definition 8.

A Latin square L of side n (or order n) is an $n \times n$ array in which each cell $L(a, b)$ contains a single symbol from an n-set S, such that each symbol occurs exactly once in each row and each column.

Definition 9.

Two Latin squares L and $L^{\prime }$ of the same order are orthogonal if $L(a, b) = L(c, d)$ and $L^{\prime }(a, b)=L^{\prime }(c, d)$ imply $a = c$ and $b = d$. A set of Latin squares $L_1, L_2, \dots , L_m$ is mutually orthogonal (or is a set of m MOLS) if for all 1 $\le i < j \le m$, $L_i$ and $L_j$ are orthogonal.

For a given n, it is common to use the notation $N\left(n\right)$ to denote the maximum number of MOLS of order n, and we say that a set of MOLS of order n is optimal if no larger set of MOLS of order n is known. Some useful results (originally from [84] and also stated in [1] [III.3.1]) are given in Theorem 4.

Theorem 4.

1. 

$N(n \times m) \ge min\left\{N\right(n), N(m\left)\right\}$.

2. 

If $n = p^e$, for prime p and some $e > 0$, then $N\left(n\right) = n-1$.

3. 

If $n = p_1^{e_1}{p}_2^{e_2}\dots p_k^{e_k}$ where each $p_i$ is a prime, then $N\left(n\right)\ge min\{p_i^{e_i}-1|i\in \{1, 2, \dots , k\}\}$.

An original comprehensive list of the then-greatest-known lower bounds on the number of MOLS of side n, where $n \le$ 10,000, was given in 1979 in [85]. The table was updated in 1996 in [86] [Table II.2.72] and again in 2007 in [1] [Table III.3.87].

Recently, the list was updated for $n \le 500$ in [87]. There, a table listing the sources of the constructions for optimal sets of MOLS was provided for each n, and a few corrections to published constructions were given. In

Table 1. Sources for MOLS constructions for $2 \le n \le 100$ where n is not a prime power.

n	$\mathit{N}\left(\mathit{n}\right)$ (Lower Bound)	Source
6, 10, 15, 20, 21, 22, 24, 26, 28, 30	1, 2, 4, 4, 5, 3, 7, 4, 5, 4	[88]
33, 34, 36, 38, 39, 40, 42, 44, 46	5, 4, 8, 4, 5, 7, 5, 5, 4
50, 51, 52, 55, 56, 62, 75, 80	6, 5, 5, 6, 7, 5, 7, 9
12	5	[89]
14	4	[90]
18, 60	5, 5	[91]
35, 48, 63	6, 10, 8	[92]
45, 54, 96	6, 8, 10	[93]
57, 69, 70, 74, 78, 84, 90	7, 6, 6, 5, 6, 6, 6	[94]
58, 66, 68	all 5	[46]
65	7	[95]
72, 77, 88, 99	7, 6, 7, 8	Theorem 4
76	6	[96]
82, 100	both 8	[97]
85, 86, 87, 92, 93, 94, 95, 98	all 6	[46]
91	7	[85]

, we provide similar information but for n up to 100 only. By convention, N$\left(n\right)$ for $n\in \{0, 1\}$ is said to be ∞, so these cases are omitted. In addition, in light of Theorem 4, we do not consider cases where n is a prime power.

Further constructions (for $n > 100$) can be found in [85,87,98,99,100,101,102,103,104].

A type of RGDD that is particularly useful for our Social Golfer solutions (as discussed in Section 2) is a resolvable transversal design. These are straightforward to construct from a suitable set of MOLS and will provide a ready source for solutions.

Definition 10.

A transversal design of order n and block size k, denoted TD$(k,n)$, is a triple $(V,G,B)$, where V is a set of $kn$ elements (or points); G is a partition of V into k classes, called groups, each of size n; B is a collection of k-subsets of V, called blocks; and every unordered pair of elements from V is contained in one group, or one block, but not both. If the blocks can be resolved into n rounds of n blocks, the design is said to be a resolvable transversal design, RTD$(k,n)$. An RTD$(k,n)$ can be obtained by removing one of the groups from a TD$(k+1, n)$.

Lemma 8.

The existence of a TD$(k,n)$ is equivalent to the existence of $k-2$ MOLS of order n. Similarly, the existence of an RTD$(k,n)$ is equivalent to the existence of $k-1$ MOLS of order n.

  Some useful RGDDs are given in Lemma 9. The first result is from [105], and the second follows from a generalisation of a theorem of a result due to Shen Hao [74] [Theorem 2]. The 9-RGDD is from [106] and is constructed by removing three parallel blocks from a 12-GDD of type $3^{45}$.

Lemma 9.

There exist the following:

1. 

A 5-RGDD of type $g^6$ if and only if $g\equiv 0 (mod 20)$;

2. 

A 5-RGDD of type $4^{30}$;

3. 

A 9-RGDD of type $3^{33}$.

When there are insufficient MOLS to construct an RTD$(k,n)$, we can still create some parallel rounds by removing a group from an incomplete transversal design, ITD. An ITD is a particular type of incomplete GDD (IGDD), which is a group divisible design for which there is a subset of points, no two of which appear together in any block (although subsets of them may appear together in the groups).

Definition 11.

An incomplete transversal design ITD$(n_1,n_2;k)$ is a tuple $(V,G,B,H)$, where V is a set of $k{n}_1$ points; G is a partition of V into k groups, each of size $n_1$; B is a collection of blocks of size k; and H is a set of $k{n}_2$ points, called a hole. Every unordered pair of elements from V is contained in a block or a group (but not both), or in the hole. Pairs of points in H may appear in a group but not in a block.

Incomplete transversal designs can be constructed via a variety of means both direct and recursive. An ITD$(n_1,n_2;k)$ can be thought of as a (hypothetical) TD$(k,n_1)$ with a (hypothetical) TD$(k,n_2)$ removed. However, often an ITD$(n_1,n_2;k)$ exists when either or both of the transversal designs TD$(k,n_1)$ and TD$(k,n_2)$ do not exist.

As an example, an ITD$(10,2;6)$ exists [107], although neither a TD$(6,10)$ nor a TD$(6,2)$ exist.

Example ITDs for $k=5$, $k=6$, and $k=7$ can be found in [97,108,109,110,111,112,113]. A number of general recursive constructions from [111,114] are summarised in [8].

Lemma 10.

For an ITD$(n_1,n_2;k)$ to exist, we must have it that $n_1\ge (k-1)n_2$.

By removing the points from a group of an ITD$(n_1,n_2;k+1)$, we achieve an incomplete group divisible design on $n_{1}k$ points with blocks of size k. There are $(n_1-n_2)$ complete parallel rounds (one for each of the removed points that are not in the hole) and $n_2$ partial parallel rounds (one for each of the removed points that are in the hole).

Example 1 follows from the fact that there is an ITD$(6,10;2)$ [107] and an ITD$(6,22;3)$ [97].

Example 1.

There exists an incomplete group divisible design on 50 points with blocks of size 5 and eight disjoint complete parallel classes. Similarly, there is an incomplete group divisible design on 110 points with blocks of size 5 and 19 disjoint complete parallel classes.

We refer to the design that consists of the blocks and groups formed by removing a group from an ITD$(n_1,n_2;k+1)$ as an RITD$(n_1,n_2;k)$.

If we view the groups of a group divisible design as blocks, we can think of an RGDD as a resolvable design for which all of the blocks (apart from those in one parallel class) have block size $k_1$ and the blocks in one parallel class have size $k_2$ (i.e., the group size). A similar structure is a uniformly resolvable pairwise balanced design ($URD$) [115]. In this case, there may be multiple block sizes, but all blocks in each parallel class have the same size.

Definition 12.

A uniformly resolvable design (URD) is a PBD whose blocks can be resolved into parallel classes in such a way that all blocks in a given parallel class have the same size.

We refer to a URD with v points and two block sizes, $k_1$ and $k_2$, where $k_1<k_2$ as a URD$(\{k_1,k_2\};v)$. The number of parallel classes of blocks of size $k_1$ and $k_2$ are denoted by $r_{k_1}$ and $r_{k_2}$, respectively. Some URDs with two small block sizes have been studied by various authors, for example, for $(k_1,k_2)=(2,3)$ [115], $(2,4)$ [83,116], $(3,4)$ [80,117,118], and $(3,5)$ [119].

Some URDs that will be useful for our allocations are those for which $v\equiv 8 (mod 12)$ and no 4-RGDD of type $2^{v/2}$ is known (see Lemma 7). The results in Lemma 11 are from [80,83] (for $n = 100$).

Lemma 11.

There exists a URD$\left(\right\{2, 4\}; 2n)$ with $r_2 = 4$ and $r_4 = (2n-5)/3$ for $n\in \{46, 70, 82, 94, 100, 118, 130, 202, 214\}$.

Example 2 is from the published best-known Social Golfer solution for $v = 30$ and $k = 5$ [30,33] (the rounds of blocks of size 2 can easily be found manually).

Example 2.

There exists a $URD\left(\right\{2, 5\}; 30)$ with $r_2 = 5$ and $r_5 = 6$.

4. (Incomplete) Transversal Design Constructions: (Incomplete) Orthogonal Arrays, Difference Matrices, and Quasi-Difference Matrices

In this section, we describe some novel constructions to create parallel classes of blocks based on difference matrices and quasi-difference matrices. These will be used for our SGA solutions in Section 7. Most of the definitions in this section are (or are similar to those) from [1] [III.3.2, III.4.1, VI.17.1 and VI.17.44] unless stated otherwise. A structure that is closely related to a transversal design is an orthogonal array (OA):

Definition 13.

An orthogonal array OA$(k,s)$ is a $k \times s^2$ array with entries from an s-set having the property that in any two rows, each (ordered) pair of symbols from S occurs exactly once.

Existence of an orthogonal array OA$(k,n)$ is equivalent to the existence of a set of $k-2$ MOLS$\left(n\right)$ or a TD$(k,n)$. See [1] [Remark III.3.10].

Lemma 12.

Let A be an OA$(k,n)$ on the n symbols in X. On $V = X \times \{0, \dots , k-1\}$ (of size $kn$), form a set B of k-sets as follows. For $0 \le j < n^2$, include $(a_{i, j}, i): 0 \le i < k\}$ in b. Then, let G be the partition of V whose classes are $\{X \times \{i\}: 0 \le i < k\}$. Then, $(V,G,B)$ is a TD$(k,n)$. This process can be reversed to recover an OA$(k,n)$ from a TD$(k,n)$.

Orthogonal arrays (and thus, MOLS) are often constructed from difference matrices. First, we define a difference matrix:

Definition 14.

Let T be an abelian group of order t. A $(t,k;\lambda )$ difference matrix (or DM) over T is a $k\times t\lambda$ matrix $D = \left(d_{x,y}\right)$ with entries from T such that for each $i, j$ satisfying $0 \le i < j < k$, the multiset $\{d_{i,l}-d_{j,l}: 0 \le l < t\lambda \}$ (the difference list) contains every element of T λ times.

Example 3.

Six MOLSs of order 35 were given in [92]. These were obtained by using permutation codes. Later, Ingo Janiszczak noted that this construction gives a set of six MOLS$\left(35\right)$ that can also be obtained more simply from the following $(35, 7; 1)$ difference matrix. Let $D_1$ and $D_3$ be, respectively, the following $7 \times 17$ and $7 \times 1$ arrays, and then let $D_2$ be the array obtained by interchanging the following pairs of rows in $D_1$: 1 and 2, 3 and 4, and 5 and 6 while keeping row 0 unaltered. Then, $D = \left[D_1\right|D_2\left|D_3\right]$ is the required $(35, 7; 1)$ difference matrix.

$D_1 = \left(\begin{array}{ccccccccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 4& 21& 3& 20& 2& 19& 1& 18& 0& 17& 34& 16& 33& 15& 32& 14& 31\\ 5& 23& 6& 24& 7& 25& 8& 26& 9& 27& 10& 28& 11& 29& 12& 30& 13\\ 19& 34& 15& 1& 13& 33& 18& 12& 27& 8& 29& 6& 26& 25& 5& 20& 0\\ 3& 23& 7& 21& 9& 24& 4& 10& 30& 14& 28& 16& 31& 32& 17& 2& 22\\ 23& 7& 33& 32& 13& 21& 9& 28& 29& 2& 0& 16& 5& 24& 25& 17& 12\\ 20& 1& 10& 11& 30& 22& 34& 15& 14& 6& 8& 27& 3& 19& 18& 26& 31\end{array}\right),$$D_3=\left(\begin{array}{c}\hfill 0\\ \hfill 22\\ \hfill 22\\ \hfill 11\\ \hfill 11\\ \hfill 4\\ \hfill 4\end{array}\right).$

If S is a symmetric $v\times v$ matrix, then five MOLS$\left(v\right)$ of the form $A_1$, $A_1^T$, $A_2$, $A_2^T$, and S are called a 2-SOLSSOM$\left(v\right)$. In [120] [Theorem 4.1], it is noted that if v is odd and there exists a $(v, 7; 1)$ DM with the property that it has an order 2 automorphism that leaves one row unaltered and permutes the other six rows in pairs, then a 2-SOLSSOM$\left(v\right)$ exists (this method was used in [120] to obtain a 2-SOLSSOM$\left(55\right)$). Hence the above $(35, 7, 1)$ DM gives a new 2-SOLSSOM$\left(35\right)$. In Theorem 5, we update [120] [Theorem 4.1] for 2-SOLSSOMs of odd order:

Theorem 5.

If v is an odd positive integer, then there exists a 2-SOLSSOM$\left(v\right)$ except for $v \in \{$3, 5} and possibly for $v \in \{$15, 21, 33, 39, 45, 51, 65, 87, 123, 135}.

An OA$(k,n)$ is obtained from a $(v,k;1)$ difference matrix D by developing the columns of D over T. Difference matrices with $\lambda > 1$ can sometimes be used to obtain OAs with $\lambda = 1$. If N is an order n subgroup of an abelian group T where $v = \lambda n$, then a $(v,k;\lambda )$ difference matrix $D = \left(d_{x,y}\right)$ over T is also said to be a difference matrix over $(T,N)$ if the equality $d_{i,l}-d_{j,l} = d_{i,s} - d_{j,s}$ implies that $d_{i,l}$ and $d_{i,s}$ are from different cosets of N in T. In [90], an OA$(6, 14)$ is obtained from a $(14, 6; 2)$ difference matrix. A modified form of this difference matrix together with the corresponding four MOLSs of order 14 are displayed in [121].

As we have previously seen, we can obtain an RTD$(k,n)$ by removing the points in a single group of a TD$(k + 1, n)$. Construction 2 will be useful when attempting to add points to rounds of blocks formed by an RTD$(5, 14)$ for an SGA solution (see Section 7).

Construction 1.

The blocks of a TD$(6, 14)$ obtained from the $(14, 6, 2)$ difference matrix in [90] can be arranged in such a way that there are ten parallel classes of blocks.

Proof. 

Let D denote the $(14, 6; 2)$ difference matrix given in [90]. As noted in [90], there are ten pairs of columns of D, which, when developed through the dihedral group $D_7$, give a parallel class.    □

Construction 2.

The blocks of an RTD$(5, 14)$ obtained from the $(14, 6, 2)$ difference matrix in [90] can be arranged in a $14 \times 14$ grid (labelled A) where the blocks in each row of A form a parallel class and the blocks in each of the first $10$ columns of A form a parallel class.

Proof. 

Again let D denote the $(14, 6; 2)$ difference matrix given in [90]. D has 28 columns with entries from $Z_{14}$, labelled as $0, 1, \dots , 27$. Fourteen of these columns have first element 0, and fourteen have first element 1.

The blocks formed by expanding ten specific disjoint pairs of columns in D form a parallel class. The appropriate ten pairs of columns are given in [90]; no column lies in more than one of these pairs.

Order the first 10 columns of D so that each of the columns labelled as $0, 1, \dots , 9$ is the member of one of those 10 pairs of columns with first entry 0, and for $x = 0, 1, \dots 9$, the column labelled as $14 + x$ is the column paired with the one labelled as x. Also choose the remaining eight columns so that (for each $y = 0, 1, 2, 3)$ the one labelled as $10 + y$ has first element 0 and the one labelled as $24 + y$ has first element 1.

An RTD$(5, 14)$ can be obtained from D. The groups in this RTD are defined as $H_s =$$\left\{\right(1, s), (2, s), \dots , (13, s\left)\right\}$ for $1 \le s \le 5$.

Now, for each column $c_j = {(c_{0,j}, c_{1,j}, \dots , c_{5,j})}^T$$(0\le j\le 27)$ in D, we construct a block $B_j = \{(c_{1,j}, 1), (c_{2,j}, 2), c_{3,j}, 3), (c_{4,j}, 4), (c_{5,j}, 5)\}$. Note that the values $c_{0,j}$, which all equal 0 or 1, do not appear in the entries of $B_j$.

Let A be the required $14 \times 14$ grid. Label the rows and columns of A as $0, 1, \dots 13$. For $i = 0, 1, \dots , 13$, place $B_i$ and $B_{i+14}$, respectively, in the $(i, 0)$ and $(i, 1)$ cells of A.

Finally, for $j = 0, 1, \dots , 27$ and $z = 2, 4, \dots , 12$, let $B_j+z=$ $\{(c_{1,j} + z,1),$$(c_{2,j}+z, 2),$$(c_{3, j} + z, 3),$$(c_{4, j} + z, 4),$$(c_{5, j} + z, 5)\}$. Here, the additions $c_{s,j} + z$ are done within $Z_{14}$. For $i = 0, 1, \dots , 13$, place $B_i+z$ and $B_{14+i}+z$, respectively, in the $(i,z)$ and $(i,z+1)$ cells of A. This gives the required grid.    □

We can extend the concept of orthogonal arrays and difference matrices to incomplete orthogonal arrays and incomplete difference matrices (known as quasi-difference matrices). The following definition is from [97]:

Definition 15.

An IOA$(k,n;h)$ (which we call A) is a $k \times (n^2 - h^2)$ array over a set X of symbols of size n. There is an h-set $Y \subset X$ such that for every pair of distinct elements $i,j\in X$ and every pair of distinct rows $s,t$ in A, there is a unique column u for which $A_{s,u} = i$ and $A_{t,u} = j$ unless both i and j lie in Y, in which case, no such column exists.

Existence of an IOA$(k,n;h)$ is equivalent to the existence of an ITD$(n,h;k)$. It follows from Lemma 10 that the existence of an IOA$(k,n;h)$ implies that $(k-1)h\le n$.

Incomplete orthogonal arrays can be constructed from quasi-difference matrices [110,122].

Definition 16.

Let G be an abelian group of order n. An $(n,k;\lambda ,\mu ;u)$-quasi-difference matrix (QDM) is a matrix $Q=\left(q_{ij}\right)$ with k rows and $\lambda (n-1 + 2u) + \mu$ columns, where each entry is either empty (usually denoted by −) or contains a single element of G. Each row contains exactly $\lambda u$ empty entries, and each column contains at most one empty entry. Furthermore, for each $0 \le i < j < k$, the multiset

$$\{q_{il}-q_{jl}:0 \le l < \lambda (n - 1 + 2u)+\mu , \mathrm{with} q_{il} \mathrm{and} q_{jl} \mathrm{not} \mathrm{empty}\}$$

contains every nonzero element of G λ times and contains 0 μ times.

The following lemma states the link between a QDM and an incomplete orthogonal array and is proved by following the construction in [1] [Construction VI.17.47].

Lemma 13.

If an $(n,k;\lambda ,\mu ;u)$-QDM exists and $\mu \le \lambda$, then an ${ITD}_{\lambda }(k,n+u;u)$ exists.

Construction 3 is also useful when attempting to add points to rounds of blocks formed by a QDM for an SGA solution (see Section 7).

Construction 3.

If an $(n,k+1;1,\mu ; u)$-QDM exists and $\mu \le 1$, then we can construct an ITD$(k,n+u;u)$ and put its blocks in an $n \times (n+u)$ array, where the blocks in each row of blocks form a parallel class and $n - (k - 1)u$ of the columns consist of a parallel class of blocks.

Proof. 

The proof is similar to that of Construction 2 and follows the steps of the ITD construction described in [1] [Construction VI.17.47]. After adding the infinite points and $1 - \mu$ columns of zeros, there are $n + 2u$ columns of which u contain an infinite point in the first position, and $k \times u$ contain an infinite point in another position. Reorder the columns so that those containing an infinite point in position i, for $0 \le i \le k$, are $\{c_j:iu \le j < (i+1)u\}$ and those that contain no infinite points are $\{c_j:(k+1)u \le j < n+2u\}$. Call the resulting matrix QDM′ and construct the ITD$(k + 1, n + u, u)$. The groups are $H_j = ((0,j), (1,j), \dots , (n+u-1,j))$ for $0 \le j \le k$. Each of the columns $c_j$ of QDM′ leads to n blocks (by adding elements of G); thus, $c_j = {(c_{0,j}, c_{1,j},\dots ,c_{k,j})}^T$ leads to the set of blocks

$$B_j = \{b_j^g:b_j^g=((c_{0,j} + g,0), (c_{1,j} + g, 1),\dots ,(c_{k,j}+g, k)), g\in G\}$$

Place the blocks from the ITD derived from the last $n + u$ columns of QDM′ in an $n \times (n+u)$ array so that the block arising from $c_j$ with first element $(i,0)$ is in position $(i,(j-u\left)\right)$ in the array.

Removing elements of group $H_0$ from each block, the blocks in each row form a parallel class. In addition, all of the columns in the array that are not formed from columns QDM′ that contain infinite points consist of parallel blocks (i.e., the last $n - \left(\right(k - 1\left)u\right)$ columns).    □

Construction 4.

If an $(n,k+1;1,\mu ; u)$-QDM exists and $\mu \le 1$, then we can construct an ITD$(k,n+u;u)$ and put its blocks in an $(n+1) \times (n+u)$ array of blocks of size k where each row of blocks is a parallel class and u of the columns consist of a parallel class of blocks.

Proof. 

Construct the first n rows as the proof of Construction 3. A further round, from the blocks of the TD containing $({\infty }_1, 0)$, should be added, with the u such blocks that are from filling the hole of the IOA placed in columns $k\ast u, (k\ast u)+1, \dots , \left(\right(k+1)\ast u) - 1$. Those u columns (of size $n + 1$) will consist of parallel blocks.    □

Definition 17.

Let $q = mt + 1$ (where $m,t$ are integers) be a prime power, and let ω be a primitive element in the finite field $F_q$. A V$(m,t)$ vector is a vector $(a_1, \dots , a_{m+1})$ for which, for each $k\in \{1, 2, \dots , m+1\}$, the differences $\{a_{i+k}-a_i:1\le i\le m+1,i+k\ne m+2\}$ represent the m cyclotomic classes of $F_{mt+1}$. Here, the subscripts $i+k$ are calculated $(mod m+2)$. In other words, for fixed k, if $a_{i+k} - a_i = {\omega }^{mx+\alpha }$ and $a_{j+k} - a_j={\omega }^{my+\beta }$ where $0\le \alpha , \beta \le m-1$ and $i\ne j$, then $\alpha \ne \beta$.

Remark 3.

A QDM can be obtained from a $V(m,t)$ vector ([1] [Construction VI.17.49], [108]).

5. Allocations: Constructions and Examples

We now return to optimal solutions to the SGP as stated in Definition 1. The problem is to create as many parallel rounds of $v/k$ blocks of size k as possible. We refer to a set of rounds of $v/k$ blocks of size k as a $(v,k)$ allocation. If there are r rounds, we refer to an r-round $(v,k)$ allocation, and if this is optimal, we refer to an optimal r-round $(v,k)$ allocation (or, more generally, as an optimal allocation). An optimal SGP solution is given by an optimal allocation, so, henceforth, we will refer to allocations rather than SGP solutions.

Remark 4.

We will refer to the number of rounds in an optimal $(v,k)$ allocation as optR$(v,k)$.

The following construction, originally due to Sharma and Das [13], is presented in [25] and provides a lower bound for r. We refer to this construction as MOLRs$(k,n)$. Note that, given a set of blocks B on a set of points V, we refer to an unused clique in B as a set of points C for which no pairs of points in C appear in any block of B. Similarly, we refer to a set of unused cliques for an allocation as a set of unused cliques in the blocks of the allocation.

Construction 5.

If there exist g MOLS$\left(n\right)$ where $n=v/k$, then there is a $(g+1)$-round $(v,k)$ allocation that has k unused cliques. These cliques are

$$C_i = \{(n\ast i)+j:0 \le j < n\}, 0 \le i < k$$

If k divides n, then we can add further rounds from the cliques, and there is an optR$(n,k)+g+1$ round $(v,k)$ allocation. See Remark 4 for the definition of optR.

Consider the g rectangles $L_{\alpha }$, $\alpha \in \{0,1,\dots ,g-1\}$ formed by taking the first k rows of the g MOLS$\left(n\right)$. These are mutually orthogonal Latin rectangles (MOLRs) as defined in [25]. Let $R_t = \{b_{t,j}, j=0, 1, \dots , n-1\}$ denote the t’th round of blocks, for $t=0, 1, \dots , g$.

Construct a $k \times n$ array G from the elements $0, 1, \dots , v-1$ as shown in Figure 4.

The first round of blocks, $R_0$, consists of the columns of G. Subsequent rounds, $R_t$, for $1 \le t \le g$, are formed by superimposing a Latin rectangle, $L_{t - 1}$, on G to form an array of pairs $G^t$, where, for $0 \le i < k$, $0 \le j < n$, $G^t\left[i\right]\left[j\right] = (G\left[i\right]\left[j\right],L_{t-1}\left[i\right]\left[j\right])$. Then, for $0 \le j < n$, $b_{t,j}$ is the list of all values $val$ for which $(val,j)$ is an element of $G^t$.

The rows of G form a set of unused cliques of size n. If k divides n, they can be broken up to form optR$(n,k)$ extra rounds of blocks (again, refer to Remark 4 for the definition of optR).

Example 4.

For $v = 36$ and $k = 6$, $n = 6$, since there is 1 MOLS$\left(6\right)$ (i.e., no pair of MOLS$\left(6\right)$ exists) and k divides n; therefore, we can construct three rounds using MOLRs$(6, 6)$.

Array G, the single Latin rectangle formed from the first six rows of the Latin square of order 6 (i.e., the Latin square itself), $L_0$, and $G^1$ formed by superimposing G and $L_0$ are shown in Figure 5.

The three rounds obtained via the Sharma and Das construction are shown in Figure 6.

Construction 6.

Rounds from starter blocks. For some $(v,k)$, the optimal allocation is obtained by expanding a set of r starter blocks. Let $g = v/k$. If we have a set of r starter blocks that have the property that

1. 

Each starter block contains 0, and none of the differences within any block is a multiple of k,

2. 

No two starter blocks intersect at any point other than 0,

3. 

For any two starter blocks $a = \{a_i: 0 \le i < k\}$ and $b = \{b_j: 0 \le j < k\}$, if there are two pairs $(a_{i_1}, b_{j_1})$ and $(a_{i_2}, b_{j_2})$ where $a_{i_1}, a_{i_2}\in a$ and $b_{j_1}, b_{j_2}\in b$ and the differences $(a_{i_1}-b_{j_1})$ and $(a_{i_2}-b_{j_2})$ are equal modulo v, then the difference is not a multiple of k,

then we can construct a set of r parallel rounds with k unused cliques of size $(v/k)$. If $n = v/k$ and k divides n, then we can add a further optR$(n,k)$ rounds from the cliques.

• The rounds are generated by adding $k\ast m$ to each starter block, for $0 \le m \le g - 1$. Each round contains distinct points (this follows from condition 1). No pair is contained in more than one block. Suppose for a contradiction that pair $(p_0, p_1)$ was to appear in two blocks, $a^{\prime }$ and $b^{\prime }$ say, derived from starter blocks a and b. Then $a^{\prime } = a+k{m}_1$ and $b^{\prime } = b+k{m}_2$ for some values $0 \le m_1 , m_2 < g$, and $(p_1, p_2) = (a_{i_1} + k{m}_1, a_{i_2} + k{m}_1)$ and $(b_{j_1} + k{m}_2, b_{j_2} + k{m}_2)$. But this contradicts condition 3. The cliques in each case are on the sets of points $S_i = \{i + jk: 0 \le j < (v/k)\}$ for $0 \le i < k$.

We refer to Construction 6 as ownSG$(v,k)$ and to the cliques as groups.

Example 5.

A seven-round $(60, 6)$ allocation can be obtained from a set of seven starter blocks.

Table 2. Pairs $(v,k)$ for which ownSG$(v,k)$ is an optimal r-round allocation or is used as a superior design for SGA solutions.

$\mathit{v},\mathit{k}$	r	$\mathit{v},\mathit{k}$	r	$\mathit{v},\mathit{k}$	r	$\mathit{v},\mathit{k}$	r	$\mathit{v},\mathit{k}$	r
$60,6$	7	$70,7$	7	$80,8$	5	$84,7$	7	$90,6$	10
$90,9$	5	$96,8$	6	$98,7$	8	$105,7$	9	$112,8$	8
$120,6$	13	$126,7$	10	$126,9$	7	$132,6$	14	$135,9$	7
$156,6$	15

presents values $(v.k)$ for cases where an r-round ownSG$(v,k)$ allocation is an optimal allocation. Starter blocks for each of these cases are given in Appendix A. The only case for which k divides $v/k = n$ is $(v,k) = (98, 7)$. There is one extra round for this example. We also note that ownSG$(156, 6)$ does not give an optimal $(156, 6)$ allocation (an RBIBD(156, 6) exists, by Theorem 2). However, ownSG$(156, 6)$ is used as a superior design (see Section 7.1) for some of our SGA allocations.

6. Optimal (v, k) Allocations

A maximal $(v,k)$ allocation is one for which no allocation with more rounds is possible. Combinatorially, the maximum number of rounds possible in a $(v,k)$ allocation is $r^{\prime }(v,k)=\lfloor \frac{(v - 1)}{(k-1)}\rfloor$. However, in some cases, non-existance results can show that, for a given instance, no allocation achieving this round is possible. We, therefore, define a maximal allocation thus (and note that a maximal allocation is clearly optimal).

Definition 18.

Let $Ma{x}_R(v,k)$ be $r - 1$, where r is the smallest integer for which an r-round $(v,k)$ allocation has been shown to not exist. Note that $Ma{x}_R(v,k) \le Ma{x}_C(v,k)$ where $Ma{x}_C(v,k)=\lfloor \frac{(v - 1)}{(k - 1)}\rfloor$. An r-round $(v,k)$-allocation with $r = Ma{x}_R(v,k)$ is said to be maximal.

We will show that when $v\equiv 0 (mod 3)$ and $k = 3$, a maximal allocation is always possible. For $v\equiv 0 (mod 4)$ and $k = 4$, a maximal allocation is almost always possible, and in the few cases where it is not known, an optimal allocation close to maximal exists. We also discuss cases where a maximal allocation is possible for $v\equiv 0 (mod 5)$ and $k = 5$ and for $v\equiv 0 (mod k)$ and $k \ge 6$.

6.1. Maximal Solutions When All Blocks Have Size 3

For an allocation to exist, we must have $v\equiv 0 (mod 3)$. Hence, either $v\equiv 3 (mod 6)$ or $v\equiv 0 (mod 6)$.

If $v\equiv 3 (mod 6)$, then we can use the blocks of a RBIBD$(v , 3)$ (see Definition 5), i.e., a Kirkman triple system on v points (KTS$\left(v\right)$), as our allocation. Since a KTS$\left(v\right)$ has $(v - 1)/2$ classes, the allocation is maximal. For every $v\equiv 3 (mod 6)$, there is a KTS$\left(v\right)$ [43], so a maximal allocation exists in all cases.

If $v\equiv 0 (mod 6)$, then we can use the blocks of a 3-RGDD of type $2^{{\displaystyle \frac{v}{2}}}$ (see Definition 7), i.e., a nearly Kirkman triple system on v points (NKTS$\left(v\right)$), as our allocation. Since NKTS$\left(v\right)$ has $(v-1)/2$ classes, the allocation is maximal. By Lemma 5, an NKTS$\left(v\right)$ exists for every $v\equiv 3 (mod 6)$ apart from $v=6$ or $v=12$, and a maximal solution exists for all $v\equiv 0 (mod 6)$, where $v\ge 18$.

6.2. Maximal Solutions When All Blocks Have Size 4

Since $v\equiv 0 (mod 4)$, either $v\equiv 4 (mod 12)$, $v\equiv 0 (mod 12)$, or $v\equiv 8 (mod 12)$. It has been previously observed in [10] that a resolvable maximum packing is equivalent to an RBIBD$(v,4)$, a 4-RGDD with groups of size 3, and a 4-RGDD with groups of size 2 respectively. We consider these cases below.

If $v\equiv 4 (mod 12)$, we can use the blocks of an RBIBD$(v,4)$ for a maximal allocation. For every $v\equiv 4 (mod12)$, there is an RBIBD$(v,4)$ [49], so a maximal allocation exists for all cases.

If $v\equiv 0 (mod 12)$, then the number of rounds in a maximal allocation is $(v - 3)/3$, which can be achieved by using the rounds of blocks of a 4-RGDD of type $3^{{\displaystyle \frac{v}{3}}}$. By Lemma 6, such an RGDD exists for all $v \ge 24$, so a maximal allocation exists for all cases $v \ge 24$.

If $v \equiv 8 (mod 12)$, then a maximal allocation can be achieved by using the rounds of blocks of a 4-RGDD of type $2^{{\displaystyle \frac{v}{2}}}$ if such an RGDD exists. By Lemma 7, there exists such an RGDD for all $v = 2u$, except when $u \in \{4, 10\}$ and possibly when $u\in \{46, 70, 82, 94, 100, 118, 130, 202, 214\}$.

For $u = 10$, we can achieve an optimal allocation by using the rounds of blocks of an RTD$(4, 5)$. As no 4-RGDD of type $2^{10}$ exists, this solution is maximal (by Definition 18). For the remaining possible exceptions, by Lemma 11, there exists a $URD\left(\right\{2, 4\}; v)$ with $r_2=4$ and $r_4 = (v-5)/3$. In all of these cases, the number of rounds is one less than that for a maximal allocation.

6.3. Maximal and Optimal Solutions When All Blocks Have Size 5

Since $v\equiv 0 (mod 5)$, $v\equiv 0$, 5, 10 or $15 (mod 20)$.

If $v\equiv 5 (mod 20)$, we can use the blocks of an RBIBD$(v, 5)$ when it exists for a maximal allocation. An RBIBD$(v, 5)$ exists in this case unless $v\in \{45, 345, 465, 645\}$ (see Lemma 3), so a maximal solution exists for all other cases.

When $v\in \{45, 345, 465, 645\}$, as there is no RBIBD$(v, 5)$, $Ma{x}_R(v, k) = Ma{x}_C(v, k)-1$ (see Definition 18). We can use an RTD$(5, v/5)$ to obtain an allocation. The allocation is not maximal in any case; indeed for the larger values, since there are only $v/5$ rounds, this is far from the case. However, currently no better solution is known.

For the remaining cases, ideally we would use a 5-RGDD with small group size; however, there are not many known examples of these. We can always use an RTD$(5, v/5)$ to obtain an allocation with $r = v/5$ rounds, except when $N (v/5) < 4$ (i.e., when $v\in \{30, 50, 110\}$).

When $v = 30$, there is a solution with six rounds, namely, the rounds of blocks of size 5 of a URD$\left(\right\{2, 5); 30)$ (see Example 2).

When $v = 50$ or $v = 110$, we use the 8 (19) parallel rounds of blocks of size 5 of the incomplete group divisible design on 50 (110) points given in Example 1.

If $v\equiv 0 (mod 20)$, we can use the $v/5$ rounds of an RTD$(5, v/5)$ if there are a sufficient number of MOLS.

When $v\equiv 0 (mod 120)$, a better solution is to use the 5-RGDD of type $g^6$ from Lemma 9, which has $5v/24$ rounds. Since the group size is divisible by 5 in this case, we can use Lemma 14 to increase the number of rounds further.

Lemma 14.

If g is divisible by k, then a uniform k-RGDD with groups of size g gives an allocation with optR$(g, k) + \left(\right(v-g)/(k-1\left)\right)$ rounds.

When $v = 120$, a better solution still is to use the blocks of the 5-RGDD of type $4^{30}$ in Lemma 9 (which has 29 rounds).

6.4. Maximal Solutions When All Blocks Have Size k, k $\ge 6$

In many cases where $k \ge 6$, it is still possible to use an RBIBD$(v, k)$ to achieve an optimal solution. An RTD$(k, v/k)$ (with the possibility of extra rounds via Lemma 14) can give solutions that are close to maximal, providing $v/k$ is small or divisible by k.

6.5. Optimal Solutions When v $\le 150$ and All Blocks Have Size k, Where k is at Least 3

The choice of method to generate optimal $(v, k)$ allocations is governed by Algorithms 1 and 2. Note that for brevity, we assume that v is divisible by k and that the input for each is v and k. As discussed in Section 6, when $k = 3$, the solution is guaranteed to be maximal, and when $k = 4$, the solution is maximal in all but nine cases (for which the solution gives one less round than a maximal solution). In all other cases, the presented solution is optimal (and in some cases, maximal, see Section 8). Note that the algorithm is guaranteed to provide an optimal solution for $v \le 150$. For $v > 150$, it will be necessary to expand the algorithm to cover additional corner cases.

Our results for $12 \le v \le 150$, where there are at least three rounds, are given in Section 8 and in the Supplementary Materials. Note that the method in each case (as denoted in the tables) is provided as a comment in Algorithm 1. We use the shorthand RGDD$(v,k,g)$ to denote a k-RGDD with groups of size g and Todorov$\left(84\right)$ to denote the blocks from the ten parallel classes from Construction 1. If the blocks of an RTD, or an RITD, or rounds from the MOLRs$(k,n)$ or ownSG$(v,k)$ constructions provide a solution, there are k disjoint unused cliques of size n (the groups). If k divides n, then the groups themselves can be divided up to form extra rounds (in a similar way to the proof of Lemma 14).

Algorithm 1 For selecting construction for optimal allocation, v points, and single block size k

Algorithm 2 Supplement to Algorithm 1 for special cases

7. When Blocks Have Adjacent Sizes—SGA

We now consider the problem where not all blocks have the same size. This problem arises when an allocation is required for v participants, but v has no divisors greater than two. If we permitted all possible block sizes, the number of solutions would be very large. We, therefore, restrict the number of block sizes to two and insist that, for block size sets $K = (k_1, k_2)$, $k_2 = k_1 + 1$ (so the block sizes are adjacent). We refer to the problem of maximising the number of rounds in this case as the Social Golfer problem with adjacent block sizes (SGA), and, as explained in Section 2.2, in this paper, we only consider $K = (4, 5)$ and $(5, 6)$.

Full results for v up to 150 are contained in our tables in the Supplementary Materials, and some cases are presented in Section 8. For completeness, we find solutions for all v (regardless of whether v has other suitable divisors) and for all values $m_1$, $m_2$ for which $k_1{m}_1 + k_2{m}_2 = v$. For brevity, we refer to a $(v, k_1, k_2, m_1, m_2)$-allocation.

By counting pairs of points, a crude estimate for the maximum possible number of rounds for a $(v, k_1, k_2, m_1, m_2)$-allocation is

$$⌊{\displaystyle \frac{v(v-1)}{m_1{k}_1(k_1-1) + m_2{k}_2(k_2-1)}}⌋.$$

Lemma 15.

A maximal upper bound on the number of rounds for a $(v, k_1, k_2, m_1, m_2)$-allocation is $⌊{\displaystyle \frac{v(v-1)}{12m_1+20m_2}}⌋$ and $⌊{\displaystyle \frac{v(v-1)}{20m_1+30m_2}}⌋$ for $K = (k_1, k_2) = (4, 5)$ and $(5, 6)$, respectively.

Our SGA solutions are derived by either removing points from a suitable allocation with more points or one with fewer points.

Definition 19.

For values v, $k_1$, $k_2$, $m_1$, and $m_2$ where $k_2=k_1+1$ and $k_1{m}_1+k_2{m}_2 = v$, we refer to the optimal $(v + m_1, k_2)$ allocation and the optimal $(v-m_2, k_1)$ allocation as the superior $(v + m_1, k_2)$-allocation and the inferior $(v-m_2, k_1)$-allocation, respectively.

In most cases, an optimal $(v, k_1, k_2,m_1,m_2)$-allocation is obtained by removing $m_1$ points from a single unused clique in the superior $(v + m_1, k_2)$-allocation.

7.1. Removing Points from an Unused Clique in the Superior Allocation

If the superior allocation is from the blocks of an RTD$(k, n)$, then any group provides a suitable clique. If k does not divide n, the subsequent solution is denoted RTD$(k, n)-m_1$, and there are n rounds. If k divides n, optR$(n,k)$ additional rounds from the groups are included in the superior allocation, RTD$(k,n)+G($optR$(n,k))$. If $m_1=1$, we simply remove a point from the final block in the final round. The solution is denoted RTD$(k,n)+G($optR$(n,k))-1$, and there are $n+$optR$(n,k)$ rounds. If $1<m_1\le k$, we remove the final round and remove $m_1$ points of the final block of that round from the rest of the design. The solution is RTD$(k,n)+G($optR$(n,k)-1)$, and there are $n+$optR$(n,k)-1$ rounds. If $m_1>k$, remove the additional rounds, reinstate the cliques, and remove $m_1$ points from one of them. The solution is RTD$(k,n)-m_1$ and there are n rounds. A similar argument applies when the superior allocation is constructed using MOLRs$(k,n)$, ownSG$(v,k)$, or RITD$(n, n_2;k)$. If points are removed from an $RBIBD(v, k)$, we can remove up to k points from the final block. When $m_1 > 1$, we also remove all blocks from the final round. A similar argument applies when the superior allocation is Todorov$\left(84\right)$. In some cases, if the superior design does not have a large enough clique, a design that is less than optimal may be used. As an example, for an optimal $(149, 5, 6, 7, 19)$-allocation, the superior allocation is an RBIBD$(156, 6)$ from which at most six points can be removed from a clique (i.e., a block). So, in this case, an ownSG$(156, 6)$ is used as a (sub-optimal) superior allocation.

7.2. Adding Points to an Inferior Allocation

In some cases (when the superior design does not provide many rounds, e.g., if the MOLR$(k,n)$ construction was used to construct it), it is preferable to add points to the inferior design instead, if possible. This involves identifying $m_2$ columns of parallel blocks in the inferior allocation and adding a new point to every block in each of these columns. For some simple designs, it is fairly easy to do this for one additional point (e.g., to create a $(41, 5, 6, 9, 1)$-allocation from an RBIBD$(40, 4)$). However, a more general approach that can be effective for larger values of $m_2$ can be used when the inferior allocation is an RTD that is constructed from a difference matrix (DM) or a quasi-difference matrix (QDM).

Consider the RTD$(5, 14)$ constructed from a $(14, 6; 2)$ DM as described in Construction 2. The blocks can be arranged in a $14 \times 14$ grid where the blocks of each row form a parallel class and the blocks in each of the first 10 columns form a parallel class.

Since the first 10 columns of blocks each form a parallel class, we can add up to 10 infinite points, one to each block in one of these columns, and still maintain the fact that each row of blocks is a parallel class. Hence, we can construct a $(70 + m_2, 5, 6, 14-m_2, m_2)$-allocation for $1 \le m_2 \le 10$, with 14 rounds in each case.

Now consider RTD$(5, 15)$, which is constructed using a $(14, 6; 1, 0; 1)$-QDM from [1] [Theorem III.3.47]. By Construction 3, we can construct a $14 \times 15$ array of blocks of size 5 where each row of blocks is a parallel class and 10 of the columns consist of parallel blocks. By adding infinite points to the columns of parallel blocks, we can construct a $(75 + m_2, 5, 6, 15 - m_2, m_2)$-allocation for $1 \le m_2 \le 10$, with 14 rounds in each case. In fact, for $m_2 = 1$, we can obtain an extra round by Construction 4.

Since the RTD$(5, 20)$ in [1] [Lemma III.3.49] is constructed using a $(19, 6; 1, 1; 1)$-QDM, by a similar argument we can construct $(100 + m_2, 5, 6, 20 - m_2, m_2)$-allocations for $1 \le m_2 \le 15$, with 20 rounds when $m_2 = 1$ and 19 rounds otherwise.

Similarly, since the RTD$(5, 26)$ in [1] [Lemma III.3.53] is constructed from a $(21, 6; 1, 0; 5)$-QDM, we can construct a $(101, 5, 6, 25, 1)$-allocation with 22 rounds.

Note that in all cases, when we run out of parallel columns of blocks, we must resort to removing points from the superior allocation.

8. Results

In this section, we provide some of our optimal SGP and SGA allocations, for $v \le 150$. These are the best allocations (i.e., with the greatest number of rounds) currently known (or possible, when maximal). When our results are not maximal (the number of rounds achieved is less than the theoretical maximum), there may be better constructions possible. We discuss this further in Section 9.

Note that full optimal allocation results, including $k \ge 3$ for SGP, and all optimal $(v,k_1,k_2,m_1,m_2)$-allocations for SGA with $(k_1, k_2)=(4, 5)$ or $(5, 6)$ for $v \le 150$ can be found in the Supplementary Materials.

8.1. SGP

We present results for optimal $(v,k)$ allocations (i.e., optimal SGP solutions) for $v \le 150$ when $k > 3$ and $v\ge k^2$. The case $v < k^2$ is omitted since here the maximal (and optimal) value of r is always $Ma{x}_R(v,k)=1$. Also, the case $k = 3$ is omitted; here, a maximal solution for $v \ge 9$ is always obtainable using a KTS$\left(v\right)$ or an NKTS$\left(v\right)$ except when $v=12$; when $v=12$, a maximal solution with $r=4$ is obtainable using an RTD$(3, 4)$.

Table 3. Key to SGP solutions in Table 4, Table 5 and Table 6.

Solution	Description
KTS $\left(v\right)$	$(v - 1)/2$ rounds of blocks of a KTS on v points
NKTS$\left(v\right)$	$(v - 2)/2$ rounds of blocks of an NKTS on v points
RBIBD$(v,k)$	$(v - 1)/(k - 1)$ rounds of blocks of an RBIBD on v points with blocks of size k
RTD$(k,n)$	n rounds of blocks of an RTD with blocks of size k and groups of size n
RGDD$(v,k,g)$	$(v-g)/(k-1)$ rounds of blocks of an RGDD with blocks of size k and groups of size g
URD$(v,k,k_1)$	$r_1$ rounds of blocks of size $k_1$ of a URD with block sizes k and $k_1$ ($r_1$ determined by the specified design)
RITD$(n_1, n_2; k)$	$(n_1 - n_2)$ rounds of blocks of an RITD
MOLRs$(k,n)$	$N\left(n\right) + 1$ rounds from Sharma and Das construction
ownSG$(v,k)$	r rounds of blocks of size k from r starter blocks
D$(v,k)$+$G\left(t\right)$	$r+t$ rounds of blocks, r from D$(v,k)$ and t from the groups of D$(v,k)$
Todorov$\left(84\right)$	10 rounds of blocks from Construction 1

contains a summary of terms contained in our SGP tables and is fully explained below. Each of

Table 4. SGP solutions for $16 \le v \le 78$. Block sizes $k \ge 4$.

v	k	${\mathit{m}}_1$	MAX	Solution	r
16	4	4	5	RBIBD $(16,4)$	5
20	4	5	5 *	RTD$(4, 5)$	5
24	4	6	7	RGDD$(24, 4, 3)$	7
25	5	5	6	RBIBD$(25, 5)$	6
28	4	7	9	RBIBD$(28, 4)$	9
30	5	6	7	URD$\left(\right\{30, 5, 2\left\}\right)$ with $r_5 = 6,$ $r_2 = 5$	6
32	4	8	10	RGDD$(32, 4, 2)$	10
35	5	7	8	RTD$(5, 7)$	7
36	4	9	11	RGDD$(36, 4, 3)$	11
	6	6	6 *	MOLRs$(6, 6) + G\left(1\right)$	3
40	4	10	13	RBIBD$(40, 4)$	13
	5	8	9	RTD$(5, 8)$	8
42	6	7	8	RTD$(6, 7)$	7
44	4	11	14	RGDD$(44, 4, 2)$	14
45	5	9	10 *	RTD$(5, 9)$	9
48	4	12	15	RGDD$(48, 4, 3)$	15
	6	8	9	RTD$(6, 8)$	8
49	7	7	8	RBIBD$(49, 7)$	8
50	5	10	12	RITD$(10, 2; 5) + G\left(1\right)$	9
52	4	13	17	RBIBD$(52, 4)$	17
54	6	9	10	RTD$(6, 9)$	9
55	5	11	13	RTD$(5, 11)$	11
56	4	14	18	RGDD$(56, 4, 2)$	18
	7	8	9	RTD$(7, 8)$	8
60	4	15	19	RGDD$(60, 4, 3)$	19
	5	12	14	RTD$(5, 12)$	12
	6	10	11	ownSG$(60, 6)$	7
63	7	9	10	RTD$(7, 9)$	9
64	4	16	21	RBIBD$(64, 4)$	21
	8	8	9	RBIBD$(64, 8)$	9
65	5	13	16	RBIBD$(65, 5)$	16
66	6	11	13	RTD$(6, 11)$	11
68	4	17	22	RGDD$(68, 4, 2)$	22
70	5	14	17	RTD$(5, 14)$	14
	7	10	11	ownSG$(70, 7)$	7
72	4	18	23	RGDD$(72, 4, 3)$	23
	6	12	14	RTD$(6, 12) + G\left(1\right)$	13
	8	9	10	RTD$(8, 9)$	9
75	5	15	18	RTD$(5, 15) + G\left(1\right)$	16
76	4	19	25	RBIBD$(76, 4)$	25
77	7	11	12	RTD$(7, 11)$	11
78	6	13	15	RTD$(6, 13)$	13

,

Table 5. SGP solutions for $80 \le v \le 117$. Block sizes $k \ge 4$.

v	k	${\mathit{m}}_1$	MAX	Solution	r
80	4	20	26	RGDD $(80, 4, 2)$	26
	5	16	19	RTD$(5, 16)$	16
	8	10	11	ownSG$(80, 8)$	5
81	9	9	10	RBIBD$(81, 9)$	10
84	4	21	27	RGDD$(84, 4, 3)$	27
	6	14	16	Todorov$\left(84\right)$	10
	7	12	13	ownSG$(84, 7)$	7
85	5	17	21	RBIBD$(85, 5)$	21
88	4	22	29	RBIBD$(88, 4)$	29
	8	11	12	RTD$(8, 11)$	11
90	5	18	22	RTD$(5, 18)$	18
	6	15	17	ownSG$(90, 6)$	10
	9	10	11	ownSG$(90, 9)$	5
91	7	13	15	RTD$(7, 13)$	13
92	4	23	30	URD$(92, 4, 2)$ with $r_4 = 29,$ $r_2 = 5$	29
95	5	19	23	RTD$(5, 19)$	19
96	4	24	31	RGDD$(96, 4, 3)$	31
	6	16	19	RTD$(6, 16)$	16
	8	12	13	ownSG$(96, 8)$	6
98	7	14	16	ownSG$(98, 7) + G\left(1\right)$	9
99	9	11	12	RGDD$(99, 9, 3)$	12
100	4	25	33	RBIBD$(100, 4)$	33
	5	20	24	RTD$(5, 20) + G\left(1\right)$	21
	10	10	10 *	MOLRs$(10, 10) + G\left(1\right)$	4
102	6	17	20	RTD$(6, 17)$	17
104	4	26	34	RGDD$(104, 4, 2)$	34
	8	13	14	RTD$(8, 13)$	13
105	5	21	26	RBIBD$(105, 5)$	26
	7	15	17	ownSG$(105, 7)$	9
108	4	27	35	RGDD$(108, 4, 3)$	35
	6	18	21	RTD$(6, 18) + G\left(1\right)$	19
	9	12	13	MOLRs$(9, 12)$	6
110	5	22	27	RITD$(22, 3; 5)$	19
	10	11	12	RTD$(10, 11)$	11
112	4	28	37	RBIBD$(112, 4)$	37
	7	16	18	RTD$(7, 16)$	16
	8	14	15	ownSG$(112, 8)$	8
114	6	19	22	RTD$(6, 19)$	19
115	5	23	28	RTD$(5, 23)$	23
116	4	29	38	RGDD$(116, 4, 2)$	38
117	9	13	14	RTD$(9, 13)$	13

and

Table 6. Some example SGP solutions for $119 \le v \le 150$. Block sizes $k \ge 4$.

v	k	${\mathit{m}}_1$	MAX	Solution	r
119	7	17	19	RTD$(7, 17)$	17
120	4	30	39	RGDD $(120, 4, 3)$	39
	5	24	29	RGDD$(120, 5, 4)$	29
	6	20	23	ownSG$(120, 6)$	13
	8	15	17	RBIBD$(120, 8)$	17
	10	12	13	MOLRs$(10, 12)$	6
121	11	11	12	RBIBD$(121, 11)$	12
124	4	31	41	RBIBD$(124, 4)$	41
125	5	25	31	RBIBD$(125, 5)$	31
126	6	21	25	RBIBD$(126, 6)$	25
	7	18	20	ownSG$(126, 7)$	10
	9	14	15	ownSG$(126, 9)$	7
128	4	32	42	RGDD$(128, 4, 2)$	42
	8	16	18	RTD$(8, 16) + G\left(1\right)$	17
130	5	26	32	RTD$(5, 26)$	26
	10	13	14	RTD$(10, 13)$	13
132	4	33	43	RGDD$(132, 4, 3)$	43
	6	22	26	ownSG$(132, 6)$	14
	11	12	13	MOLRs$(11, 12)$	6
133	7	19	22	RTD$(7, 19)$	19
135	5	27	33	RTD$(5, 27)$	27
	9	15	16	ownSG$(135, 9)$	7
136	4	34	45	RBIBD$(136, 4)$	45
	8	17	19	RTD$(8, 17)$	17
138	6	23	27	RTD$(6, 23)$	23
140	4	35	46	URD$(140, 4, 2)$ with $r_4 = 45,$ $r_2 = 5$	45
	5	28	34	RTD$(5, 28)$	28
	7	20	23	MOLRs$(7, 20)$	5
	10	14	15	MOLRs$(10, 14)$	5
143	11	13	14	RTD$(11, 13)$	13
144	4	36	47	RGDD$(144, 4, 3)$	47
	6	24	28	RTD$(6, 24) + G\left(1\right)$	25
	8	18	20	MOLRs$(8, 18)$	6
	9	16	17	RTD$(9, 16)$	16
	12	12	13	MOLRs$(12, 12) + G\left(1\right)$	7
145	5	29	36	RBIBD$(145, 5)$	36
147	7	21	24	MOLRs$(7, 21) + G\left(1\right)$	7
148	4	37	49	RBIBD$(148, 4)$	49
150	5	30	37	RTD$(5, 30) + G\left(6\right)$	36
	6	25	29	RTD$(6, 25)$	25
	10	15	16	MOLRs$(10, 15)$	5

contains SGP solutions for an interval of values v, for $v\le 150$. For each value v, the possible block sizes k are considered: $m_1$ denotes the number of blocks of size k in each round. $MAX$ denotes the maximum possible number of rounds (i.e., $Ma{x}_R(v,k)$ as defined in Definition 18). We have indicated the few cases where this is not simply $Ma{x}_C(v,k)$ (also defined in Definition 18) using an asterisk. The number of rounds achieved using the solution provided is r.

Table 3 contains descriptions of the solutions provided (including the KTS and NKTS cases omitted here). Note that $N\left(v\right)$ denotes the (current lower bound on the) number of MOLS of order v and $n=v/k$. Note that KTS, NKTS, RBIBD, RTD, RGDD, URD, and RITD are the designs introduced in Section 3, and Todorov$\left(84\right)$ is the ten parallel rounds of blocks constructed using Construction 1. Design D$(v,k)$ represents any element of $\mathcal{D}=\{$RTD$(k,n)$, RGDD$(v,k,g)$, URD$(v,k,k_1)$, RITD$(n_1,n_2;k)$, MOLRs$(k,n)$, ownSG$(v,k)\}$.

8.2. SGA

Even for the restricted block sizes pairs $(4, 5)$ and $(5, 6)$, there are far too many solutions to be included in full here. Instead, we present a selection of the more difficult cases.

Table 7. Key to SGA solutions in Table 8.

Solution	Description
RBIBD$(v,k) - 1$	$(v - 1)/(k - 1)$ rounds of blocks of an RBIBD on v points with blocks of size k, with a single point removed.
RBIBD$(v,k) - t,B$	$(v - 1)/(k - 1) - 1$ rounds of blocks from an RBIBD on v points with blocks of size k, from which $t > 1$ points have been removed from the final block, and the final round of blocks is removed.
Todorov$\left(84\right) - 1$	10 rounds of blocks from Todorov$\left(84\right)$, with a single point removed.
Todorov$\left(84\right) - t, B$	9 rounds of blocks from Todorov$\left(84\right)$, from which $t > 1$ points have been removed from the final block, and the final round of blocks is removed.
E$(v,k) - 1$	r rounds of blocks from design E$(v,k)$ with a single point removed.
D$(v,k) - t$, ($t > 1$)	r rounds of blocks from the r rounds of design D$(v,k)$ from which t points have been removed from one of its (unused) groups.
D$\prime (v,k) - t, B$, ($t > 1)$	$r - 1$ rounds of blocks from the r rounds of design D′$(v,k)$ from which t points have been removed from the final block, and the final round of blocks is removed.
RTD$(k,n)\_D + t$	n rounds of blocks from adding t points to the n parallel rounds of the RTD$(k,n)$ constructed from a DM.
RTD$(k,n)\_Q(g,u) + t$	r rounds of blocks from adding t points to the r parallel rounds of the RTD$(k,n)$ constructed from a $(g,k + 1; 1, \mu ; u)$-QDM. Here, r is $g + 1$ if $t \le u$ and g otherwise.

contains descriptions of the solutions provided (including for the solutions of the type omitted here for completeness). In the Solution column, D′$(v,k)$ and E$(v,k)$ represent any element of sets ${\mathcal{D}}^{\prime }$ and $\mathcal{E}$ where ${\mathcal{D}}^{\prime }=\{$D$(v,k) + G\left(t\right):$D$(v,k)\in \mathcal{D}\}$, where $G\left(t\right)$ denotes that t rounds are added from the groups (or unused cliques) of the design associated with D$(v,k)$; and $\mathcal{E}=\mathcal{D}\cup {\mathcal{D}}^{\prime }$. In

Table 8. SGA solutions for $25 \le v \le 150$. Block sizes $(k_1, k_2)=(4, 5)$, or $(5, 6)$.

v	${\mathit{k}}_1, {\mathit{k}}_2$	${\mathit{m}}_1, {\mathit{m}}_2$	MAX	Solution	r
$v\in [25, 28]$	$4, 5$	$30 - v, v - 24$	7	URD$(30, 5, 2)-m_1, B$	5 for $v\in [25, 26]$, 6 otherwise
29	$4, 5$	$1, 5$	7	URD$(30, 5, 2)-m_1$	6
$v\in [32, 35]$	$5, 6$	$36 - v, v - 30$	7	MOLRs$(6, 6) + G\left(1\right) - m_1$	3
31	$5, 6$	$5, 1$	7	MOLRs$(6, 6) - m_1$	2
$v\in [42, 48]$	$4, 5$	$50 - v, v - 40$	12	RITD$(10, 2; 5) - m_1$	8
49	$4, 5$	$1, 9$	12	RITD$(10, 2; 5) + G\left(1\right) - 1$	9
$v\in [51, 59]$	$5, 6$	$60 - v, v - 50$	12 for $v\in [51, 52]$, 11 otherwise	ownSG$(60, 6) - m_1$	7
$v\in [71, 80]$	$5, 6$	$84 - v, v - 70$	17 for $v\in [71, 72]$, 16 otherwise	RTD$(5, 14)\_D + m_2$	14
$v\in [76, 85]$	$5, 6$	$90 - v, v - 75$	18 for $v\in [76, 81]$, 17 otherwise	RTD$(5, 15)\_Q(14, 1) + m_2$	15 for $v = 76$, 14 otherwise
$v\in [86, 89]$	$5, 6$	$90 - v, v - 75$	17	ownSG$(90, 6) - m_1$	10
$v\in [91, 109]$	$4, 5$	$110 - v, v - 88$	28 for $v\in [91, 94]$, 27 otherwise	RITD$(22, 3; 5) - m_1$	19
$v\in [101, 115]$	$5, 6$	$120 - v, v - 100$	24 for $v\in [101, 109]$, 23 otherwise	RTD$(5, 20)\_Q(19, 1)+m_2$	20 for $v = 101$, 19 otherwise
$v\in [111, 131]$	$5, 6$	$132 - v, v - 110$	27 for $v = 111$, 26 otherwise	ownSG$(132, 6) - m_1$	14
$v\in [116, 119]$	$5, 6$	$120 - v, v - 100$	23	ownSG$(120, 6) - m_1$	13
131	$5, 6$	$25, 1$	32	RTD$(5, 26)\_Q(21, 5) + 1$	22
$v\in [132, 149]$	$5, 6$	$156 - v, v - 130$	32 for $v = 132$, 31 otherwise	ownSG$(156, 6) - m_1$	15

we include solutions that are not obtained by removing points from an RBIBD$(v,k_2)$, Todorov$\left(84\right)$, or an RTD$(k_2,v/k_2)$ or any other RGDD (with or without additional rounds from the groups). In this case, pairs of block sizes $K = (k_1, k_2)$ (abbreviated to $k_1, k_2$) are considered, and $m_1$ and $m_2$ denote the number of blocks of size $k_1$ and $k_2$ in each round. The value of $MAX$ is determined by the bounds in Lemma 15. Note that, to ensure there are at least two rounds, we must have $m_1 + m_2 \ge k_2$.

8.3. Reproducibility and Verification

Our results can be reproduced by following the construction of the underlying design from the corresponding source (see Section 3) or from our descriptions of constructions in Section 4 and Section 5. Once the constructions have been implemented, the corresponding allocation can be generated in a few seconds, even for the larger values of v.

All of our allocations, together with verification scripts and instructions for use, can be found at our online repository at [123].

9. Conclusions

We have investigated the use of combinatorial designs to find solutions to the Social Golfer problem (SGP) and the Social Golfer problem with adjacent group sizes (SGA). Any solution with the theoretically maximum number of rounds is called a maximal solution, and solutions with the number of rounds that is the best currently known (but is not necessarily maximal) are said to be optimal.

Existing design theoretic approaches to find solutions include maximal solutions from resolvable balanced incomplete block designs; (then) optimal solutions using mutually orthogonal Latin rectangles (MOLRs); and a ten-round (maximal) solution for 32 players in groups of size 4 that was shown to be equivalent to the rounds of blocks of a resolvable group divisible design.

However, most previous research into SGP focuses on the development of efficient search algorithms to find optimal solutions (which may, or may not, be maximal). The goal in this case is to compare the performance of proposed algorithms with previous approaches in terms of the number of rounds obtained and how quickly a solution is obtained, rather than in the solutions themselves. In addition, SGA has only recently been investigated (and only for up to 50 players).

The goal of this paper is to construct (and show how to systematically select constructions for) optimal (in some cases, maximal) solutions to the SGP and SGA by leveraging resolvable design theory and related combinatorial structures, consolidating tools that are widely used in this area while extending/organising solution coverage and providing an algorithmic “construction chooser” framework.

Specifically, we have surveyed the combinatorial structures that can be used to find optimal solutions to the Social Golfer problem (SGP) and the Social Golfer problem with adjacent group sizes (SGA). We have provided a number of new constructions and an algorithm to determine which method to use to find an optimal $(v,k)$ allocations (i.e., solutions to SGP) for any pair $(v,k)$, when $k > 2$ divides v. When groups have size 3, the algorithm returns a maximal solution for any number of participants. When groups have size 4, the solution is maximal for all but nine cases. In these instances, the solution gives one fewer round than the theoretical maximum. Otherwise, the algorithm suggests a solution that is guaranteed to be optimal for $v \le 150$ (but may not address corner cases for $v > 150$). We have also shown how optimal $(v, k_1, k_2, m_1, m_2)$-allocations (i.e., optimal solutions for SGA) can be found by either adding points to an inferior allocation or removing points from an unused clique.

Selected results for $v \le 150$ and suitable block size k (or block sizes $(k_1, k_2) = \{4, 5\}$ or $\{5, 6\}$) are given in Section 8. Complete results are provided in tables in the Supplementary Materials. Our allocations can be downloaded from our BoRAT website, the link for which can be found at [32].

As our solutions are not maximal in many cases, they may indeed be improved upon in the future. In addition, our presented results only apply for $v \le 150$ (although, in many cases, they can naturally applied to larger cases). However, by providing such a systematic classification for the first time, we have laid the groundwork for future research in this area.