Solving the Kirkman’s Schoolgirl Problem in a Few Seconds

Scheduling Social Golfers with Memetic Evolutionary Programming

“…Finally, symmetries among golfers are harder to handle: they can be fully removed only by using advanced techniques for dynamic symmetry breaking (see [4] for more details).…”

“…Also, Gent and Lynce [10] have recently introduced a satisfiability (SAT) encoding for the SGP. Barnier and Brisset [4] have presented a combination of techniques to efficiently find solutions to a specific instance of SGP, the Kirkman's schoolgirl problem. Global constraints for lexicographic ordering have been proposed by Frisch et al [11], being used for breaking symmetries in the SGP.…”

“…First, the tabu list is distributed across the various weeks, which is natural since the swaps only consider golfers in the same week. The tabu component thus consists of an array tabu, where tabu[w] represents the tabu list associated with week w. Second, for a given week w, the tabu list maintains triplets a, b, i , where a and b are two golfers, and i represents the first iteration where golfers a and b can be swapped again in week w. Third, the tabu tenure, i.e., the time a pair of golfers (a, b) stays in the list, is dynamic: it is randomly generated in the interval [4,100]. In other words, each time a pair of golfers (a, b) is swapped, a random value ρ is drawn uniformly from the interval [4,100] and the pair (a, b) is tabu for the next ρ iterations.…”

“…Notice in this sense that symmetry is manifold in this problem, e.g., players can be permuted within groups, groups can be ordered arbitrarily within every week, and even the weeks themselves can be permuted. Recent developments (e.g., [4]) approach the scheduling of social golfers using innovative, elegant, but also complex, symmetry-breaking schemes.…”

Compositional Derivation of Symmetries for Constraint Satisfaction

“…Recent research has mostly focused on breaking symmetries, including general symmetry-breaking schemes (e.g., SBDS (Backofen & Will 1999;Gent & Smith 2000) and SBDD (Fahle, Schamberger, & Sellmann 2001;Focacci & Milano 2001)), their efficient implementations (e.g., (Puget 2003)), and their specialisations for specific applications (e.g., (Barnier & Brisset 2002;Puget 2002)). There has also been a tendency to abstract some of the techniques from particular applications to classes of CSPs (Van Hentenryck et al 2003) or models (Flener et al 2002).…”