Solving the Latin Square Completion Problem by Memetic Graph Coloring

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.