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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.

Cite AsGet BibTex

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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