Abstract
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.
Highlights
A LATIN square L of order n is composed of n × n grids such that each grid is filled with a number in {1, . . . , n} (n ∈ N+) and each number occurs in each row and each column exactly once
We show in the Appendix additional results on the related partial Latin square extension (PLSE) problem
By taking advantage of the particular features of Latin square graphs, we have developed a constraint propagation-based kernelization technique to preprocess the given Latin square graph to obtain a reduced graph, for which an associated list coloring problem is defined
Summary
A LATIN square L of order n is composed of n × n grids (or cells) such that each grid is filled with a number in {1, . . . , n} (n ∈ N+) and each number occurs in each row and each column exactly once. While the reference approaches can only solve a subset of the tested instances, our approach is able to solve all the instances consistently Such a performance has never been reported in the literature, demonstrating the high effectiveness of considering LSC as a graph coloring problem and using the proposed populationbased memetic algorithm to color Latin square graphs. More generally, the proposed method can be used to solve the list coloring and precoloring extension problems, which are relevant graph models both in theory and in practice [28] For these two important coloring problems, the literature offers many theoretical studies on specific graphs, we are not aware of any dedicated and effective algorithm able to handle large graphs. The Appendix reports computational results of the proposed method on the related PLSE problem
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