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

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.