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

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

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Author Details

Renee Mirka
  • Cornell University, Ithaca, NY, USA
Laura Greenstreet
  • Cornell University, Ithaca, NY, USA
Marc Grimson
  • Cornell University, Ithaca, NY, USA
Carla P. Gomes
  • Cornell University, Ithaca, NY, USA

Funding

This material is based upon work partially supported by the National Science Foundation under Awards CCF-1522054 and CCF-2007009 and by AFOSR DURIP grant FA9550-21-1-0316, AFOSR MURI grant FA9550-18-1-0136, and AFOSR grant FA9550-20-1-0421.

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Renee Mirka, Laura Greenstreet, Marc Grimson, and Carla P. Gomes. A New Approach to Finding 2 x n Partially Spatially Balanced Latin Rectangles (Short Paper). In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 280, pp. 47:1-47:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CP.2023.47

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.

Subject Classification

ACM Subject Classification
  • Applied computing → Operations research
Keywords
  • Spatially balanced Latin squares
  • partially spatially balanced Latin rectangles
  • minimum edge weight clique
  • combinatorial optimization
  • mixed integer programming
  • imbalance
  • cliques

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References

  1. Luis A Agapito, Marco Fornari, Davide Ceresoli, Andrea Ferretti, Stefano Curtarolo, and Marco Buongiorno Nardelli. Accurate tight-binding hamiltonians for two-dimensional and layered materials. Physical Review B, 93(12):125137, 2016. Google Scholar
  2. Bahram Alidaee, Fred Glover, Gary Kochenberger, and Haibo Wang. Solving the maximum edge weight clique problem via unconstrained quadratic programming. European Journal of Operational Research, 181(2):592-597, 2007. Google Scholar
  3. Egon Balas and Chang Sung Yu. Finding a maximum clique in an arbitrary graph. SIAM Journal on Computing, 15(4):1054-1068, 1986. Google Scholar
  4. Mateo Díaz, Ronan Le Bras, and Carla Gomes. In search of balance: The challenge of generating balanced latin rectangles. In Proceedings of Integration of AI and OR Techniques in Constraint Programming: 14th International Conference, pages 68-76. Springer, 2017. Google Scholar
  5. G Dijkhuizen and Ulrich Faigle. A cutting-plane approach to the edge-weighted maximal clique problem. European Journal of Operational Research, 69(1):121-130, 1993. Google Scholar
  6. Stefano Ermon, Carla Gomes, Ashish Sabharwal, and Bart Selman. Low-density parity constraints for hashing-based discrete integration. In Proceedings of International Conference on Machine Learning, pages 271-279. PMLR, 2014. Google Scholar
  7. U Faigle, R Garbe, K Heerink, and B Spieker. Lp—relaxations for the edge—weighted subclique problem. In Operations Research’93: Extended Abstracts of the 18th Symposium on Operations Research held at the University of Cologne September 1-3, 1993, pages 157-160. Springer, 1994. Google Scholar
  8. Carla Gomes and Meinolf Sellmann. Streamlined constraint reasoning. In Proceedings of Principles and Practice of Constraint Programming: 10th International Conference, pages 274-289. Springer, 2004. Google Scholar
  9. Carla Gomes, Meinolf Sellmann, Cindy Van Es, and Harold Van Es. The challenge of generating spatially balanced scientific experiment designs. In Proceedings of the International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, pages 387-394. Springer, 2004. Google Scholar
  10. Carla P Gomes, Joerg Hoffmann, Ashish Sabharwal, and Bart Selman. Short xors for model counting: from theory to practice. In Proceedings of Theory and Applications of Satisfiability Testing-SAT 2007: 10th International Conference, pages 100-106. Springer, 2007. Google Scholar
  11. Carla P Gomes, Ashish Sabharwal, and Bart Selman. Model counting: A new strategy for obtaining good bounds. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 10, pages 1597538-1597548, 2006. Google Scholar
  12. Carla P Gomes, Willem Jan van Hoeve, Ashish Sabharwal, and Bart Selman. Counting csp solutions using generalized xor constraints. In Proceedings of the AAAI Conference on Artificial Intelligence, pages 204-209, 2007. Google Scholar
  13. Seyedmohammadhossein Hosseinian, Dalila BMM Fontes, Sergiy Butenko, Marco Buongiorno Nardelli, Marco Fornari, and Stefano Curtarolo. The maximum edge weight clique problem: formulations and solution approaches. Optimization Methods and Applications: In Honor of Ivan V. Sergienko’s 80th Birthday, pages 217-237, 2017. Google Scholar
  14. Seyedmohammadhossein Hosseinian, DBMM Fontes, and Sergiy Butenko. A quadratic approach to the maximum edge weight clique problem. In XIII Global Optimization Workshop GOW, volume 16, pages 125-128, 2016. Google Scholar
  15. Marcel Hunting, Ulrich Faigle, and Walter Kern. A lagrangian relaxation approach to the edge-weighted clique problem. European Journal of Operational Research, 131(1):119-131, 2001. Google Scholar
  16. Ronan Le Bras, Carla Gomes, and Bart Selman. From streamlined combinatorial search to efficient constructive procedures. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 26-1, pages 499-506, 2012. Google Scholar
  17. Ronan Le Bras, Andrew Perrault, and Carla P Gomes. Polynomial time construction for spatially balanced latin squares. Technical report, Cornell University, 2012. Google Scholar
  18. Anuj Mehrotra. Cardinality constrained boolean quadratic polytope. Discrete Applied Mathematics, 79(1-3):137-154, 1997. Google Scholar
  19. Michael M Sørensen. New facets and a branch-and-cut algorithm for the weighted clique problem. European Journal of Operational Research, 154(1):57-70, 2004. Google Scholar
  20. Etsuji Tomita, Tatsuya Akutsu, and Tsutomu Matsunaga. Efficient algorithms for finding maximum and maximal cliques: Effective tools for bioinformatics. IntechOpen, 2011. Google Scholar
  21. HM Van Es and CL Van Es. Spatial nature of randomization and its effect on the outcome of field experiments. Agronomy Journal, 85(2):420-428, 1993. Google Scholar
  22. Pascal Van Hentenryck and Laurent Michel. Differentiable invariants. In Proceedings of Principles and Practice of Constraint Programming: 12th International Conference, pages 604-619. Springer, 2006. Google Scholar
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