The Latin square completion (LSC) problem involves completing a partially filled Latin square of order ${n}$ by assigning numbers from 1 to ${n}$ to the empty grids such that each number occurs exactly once in each row and each column. LSC has numerous applications and is, however, NP-complete. In this paper, we investigate an approach for solving LSC by converting an LSC instance to a domain-constrained Latin square graph and then solving the associated list coloring problem. To be effective, we first employ a constraint propagation-based kernelization technique to reduce the graph model and then call for a dedicated memetic algorithm to find a legal list coloring. The population-based memetic algorithm combines a problem-specific crossover operator to generate meaningful offspring solutions, an iterated tabu search procedure to improve the offspring solutions, and a distance-quality-based pool updating strategy to maintain a healthy diversity of the population. Extensive experiments on more than 1800 LSC benchmark instances in the literature show that the proposed approach can successfully solve all the instances, surpassing the state-of-the-art methods. To our knowledge, this is the first approach achieving such a performance for the considered problem. We also report computational results for the related partial Latin square extension problem.

中文翻译：

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通过模因图着色解决拉丁方完成问题

拉丁方补全 (LSC) 问题涉及通过将 1 到 ${n}$ 的数字分配给空网格来完成部分填充的 ${n}$ 拉丁方，使得每个数字在每行和每列中恰好出现一次. LSC 有很多应用，但是，它是 NP 完全的。在本文中，我们研究了一种通过将 LSC 实例转换为受域约束的拉丁方图，然后解决相关列表着色问题来解决 LSC 的方法。为了有效，我们首先采用基于约束传播的核化技术来减少图模型，然后调用专用的模因算法来找到合法的列表着色。基于群体的模因算法结合了特定于问题的交叉算子来生成有意义的后代解决方案，迭代禁忌搜索程序以改进后代解决方案，以及基于距离质量的池更新策略以保持种群的健康多样性。对文献中 1800 多个 LSC 基准实例的大量实验表明，所提出的方法可以成功解决所有实例，超越了最先进的方法。据我们所知，这是针对所考虑的问题实现这种性能的第一种方法。我们还报告了相关的部分拉丁方扩展问题的计算结果。超越最先进的方法。据我们所知，这是针对所考虑的问题实现这种性能的第一种方法。我们还报告了相关的部分拉丁方扩展问题的计算结果。超越最先进的方法。据我们所知，这是针对所考虑的问题实现这种性能的第一种方法。我们还报告了相关的部分拉丁方扩展问题的计算结果。