Abstract
Holey Latin square (HLS) is a special combinatorial design of interest to mathematicians and is helpful in the construction of many important structures in design theory. In this paper, we investigate the existence of HLSs satisfying the seven kinds of identities with automated reasoning techniques. We formulate this problem as propositional logic formulae. Since state-of-the-art SAT solvers have difficulty solving many HLS problems, we further propose a symmetry breaking method, called partially ordered HLS (POHLS), to eliminate isomorphic solutions. We have achieved the following goals through experimental evaluation. First, we have solved a dozen of open problems interested by mathematicians. Second, we identify the impact of different encodings. Third, we demonstrate the advantages of SAT solver over other FOL-based solvers. Fourth, we show that the proposed POHLS reduction can improve the efficiency of solving and find the complementarity between two types of symmetry breaking techniques.
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Notes
- 1.
L. Zhu, private communication with F. Ma, July 2020.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (NSFC) under grants No. 62132020 and No. 61972384. Feifei Ma is also supported by the Youth Innovation Promotion Association CAS under grant No. Y202034. The authors would like to thank Lie Zhu at SooChow University for suggesting the open problems and providing help.
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Liu, M. et al. (2024). Investigating the Existence of Holey Latin Squares via Satisfiability Testing. In: Liu, F., Sadanandan, A.A., Pham, D.N., Mursanto, P., Lukose, D. (eds) PRICAI 2023: Trends in Artificial Intelligence. PRICAI 2023. Lecture Notes in Computer Science(), vol 14326. Springer, Singapore. https://doi.org/10.1007/978-981-99-7022-3_38
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