In Search of Balance: The Challenge of Generating Balanced Latin Rectangles
Spatially Balanced Latin Squares are combinatorial structures of great importance for experimental design. From a computational perspective they present a challenging problem and there is a need for efficient methods to generate them. Motivated by a real-world application, we consider a natural extension to this problem, balanced Latin Rectangles. Balanced Latin Rectangles appear to be even more defiant than balanced Latin Squares, to such an extent that perfect balance may not be feasible for Latin rectangles. Nonetheless, for real applications, it is still valuable to have well balanced Latin rectangles. In this work, we study some of the properties of balanced Latin rectangles, prove the nonexistence of perfect balance for an infinite family of sizes, and present several methods to generate the most balanced solutions.
KeywordsLatin Rectangles Experimental design Local search Constraint satisfaction problem Mixed-integer programming
This work was supported by the National Science Foundation (NSF Expeditions in Computing awards for Computational Sustainability, grants CCF-1522054 and CNS-0832782, NSF Computing research infrastructure for Computational Sustainability, grant CNS-1059284).
- 1.Ermon, S., Gomes, C.P., Sabharwal, A., Selman, B.: Low-density parity constraints for hashing-based discrete integration. In: Proceedings of the 31th International Conference on Machine Learning, ICML 2014, Beijing, China, 21–26 June 2014, pp. 271–279 (2014)Google Scholar
- 3.Gent, I.P., Smith, B.M.: Symmetry breaking in constraint programming. In: Proceedings of ECAI-2000, pp. 599–603. IOS Press (2000)Google Scholar
- 6.Gomes, C.P., Sabharwal, A., Selman, B.: Model counting: a new strategy for obtaining good bounds. In: Proceedings, The Twenty-First National Conference on Artificial Intelligence and the Eighteenth Innovative Applications of Artificial Intelligence Conference, Boston, Massachusetts, USA, 16–20 July 2006, pp. 54–61 (2006)Google Scholar
- 7.Gomes, C.P., Sellmann, M.: Streamlined constraint reasoning. In: Principles and Practice of Constraint Programming - CP 2004, 10th International Conference, CP 2004, Toronto, Canada, 27 September–1 October 2004, Proceedings, pp. 274–289 (2004)Google Scholar
- 8.Gomes, C.P., van Hoeve, W.J., Sabharwal, A., Selman, B.: Counting CSP solutions using generalized XOR constraints. In: AAAI, pp. 204–209 (2007)Google Scholar
- 9.Van Hentenryck, P., Michel, L.: Differentiable invariants. In: Principles and Practice of Constraint Programming - CP 2006, 12th International Conference, CP 2006, Nantes, France, 25–29 September 2006, Proceedings, pp. 604–619 (2006)Google Scholar
- 10.Le Bras, R., Gomes, C.P., Selman, B.: From streamlined combinatorial search to efficient constructive procedures. In: AAAI (2012)Google Scholar
- 11.Le Bras, R., Perrault, A., Gomes, C.: Polynomial time construction for spatially balanced Latin squares (2012)Google Scholar
- 13.Smith, C., Gomes, C., Fernandez, C.: Streamlining local search for spatially balanced Latin squares. In: IJCAI, vol. 5, pp. 1539–1541. Citeseer (2005)Google Scholar