A New Approach to Finding 2 x n Partially Spatially Balanced Latin Rectangles (Short Paper)

Renee Mirka, Laura Greenstreet, Marc Grimson, Carla P. Gomes

Abstract: Partially spatially balanced Latin rectangles are combinatorial structures that are important for experimental design. However, it is computationally challenging to find even small optimally balanced rectangles, where previous work has not been able to prove optimality for any rectangle with a dimension above size 11. Here we introduce a graph-based encoding for the 2 × n case based on finding the minimum-cost clique of size n. This encoding inspires a new mixed-integer programming (MIP) formulation, which finds exact solutions for the 2 × 12 and 2 × 13 cases and provides improved bounds up to n = 20. Compared to three other methods, the new formulation establishes the best lower bound in all cases and establishes the best upper bound in five out of seven cases.