Investigating the Existence of Orthogonal Golf Designs via Satisfiability Testing

Abstract

A collection of n-2 idempotent symmetric quasigroups of order n is called a golf design if all the quasigroups in the collection are mutually disjoint. Two golf designs are said to be orthogonal if any idempotent symmetric quasigroup from one golf design has an orthogonal mate in the other golf design, and it is also called an orthogonal golf design (OG(n)). The existence of orthogonal golf designs is an open problem in combinatorial design theory. In this paper, we describe a method for solving some open cases using automated reasoning tools, employing both symmetry breaking and heuristic decision. The experimental results show that our method is highly efficient and it indeed allowed us to get some positive results in reasonable time. In particular, we apply state-of-the-art SAT solvers and constraint solvers to decide the non-existence of some instances, which can produce a formal proof.