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

TL;DR

This work addresses the Social Golfer Problem (SGP) and its adjacent-block-size variant (SGA) by leveraging resolvable combinatorial designs to construct optimal, repeatable round-robin allocations. It introduces an algorithmic framework that selects among RBIBD, RGDD, URD, RTD, MOLRs, and starter-block constructions (augmented by OAs, DMs, and QDMs) to maximize the number of rounds $r$ for a given $(v,k)$, with complete optimal solutions provided for $v\le 150$ and $k\ge3$. The authors extend SGP methodology to SGA by handling two adjacent block sizes $k_1$ and $k_2=k_1+1$, using removal or addition of points to transform existing designs into valid adjacent-size solutions, and they supply extensive tables and a web resource (BoRAT) for practical deployment. Overall, the paper offers a comprehensive, design-theoretic suite of constructions and an algorithmic toolkit that yields high-signal, application-ready allocations for a broad range of problem instances, including cases where divisors are scarce or where symmetry-breaking constraints are critical for tractable search.

Abstract

Resolvable combinatorial designs including Resolvable Balanced Incomplete Block Designs, Resolvable Group Divisible Designs, Uniformly Resolvable Designs and Mutually Orthogonal Latin Squares and Rectangles are used to construct optimal solutions to the Social Golfer problem (SGP) and the Social Golfer problem with adjacent group sizes (SGA). An algorithm is presented to find an optimal solution in general, and a complete set of solutions is provided for up to 150 players.

Figures (14)

• Figure 1: Maximal Social Golfer solution for v=28 and groups of size 4. There are $9$ rounds, each containing $7$ groups. Every pair of players appears exactly once.
• Figure 2: Maximal Social Golfer solution for v=24 and groups of size 4. There are $7$ rounds, each containing $6$ groups. Not all pairs of players appear.
• Figure 3: Maximal Social Golfer solution for v=22 and groups of size 4 and 3, obtained by removing players $0$ and $8$ from \ref{['fig:SGPExample3']}. Note that players are not sequentially labelled.
• Figure 4: Maximal Social Golfer solution for v=22 and groups of size 4 and 3, obtained by removing players $0$ and $8$ from \ref{['fig:SGPExample3']}, and with appropriate renaming so that players are sequentially labelled.
• Figure 5: An RBIBD$(16,4)$.

Theorems & Definitions (47)

• definition 1
• definition 2
• definition 3
• lemma 1
• definition 4
• definition 5
• Example 1
• Example 2
• lemma 2
• lemma 3