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.

A deep learning guided memetic framework for graph coloring problems

“…Starting from a solution constructed using greedy methods, local search improves the current solution by considering best moves in a given neighborhood. To be effective, local search heuristics usually incorporate mechanisms to escape local optima based on tabu lists [3,12] or perturbation strategies [16,27]. However for very difficult instances of graph coloring, this is often not enough to find the global optimum as the search may be restricted to a single region of the search space.…”

A massively parallel evolutionary algorithm for the partial Latin square extension problem

“…The PLSE is to color as many uncolored vertices as possibles such that two colored vertices do not share the same color. Based on this observation, the authors of [15] proposed a powerful memetic algorithm for the Latin square completion problem and solved all the 1800 LSC instances introduced in [13] as well as all the 19 traditional LSC instances in the literature [11]. With some slight adaptations of their algorithm, they also reported excellent results on the 1800 PLSE instances of [13].…”

“…To sum, the two most recent studies on the PLSE [13] and the LSC [15] significantly contributed to the practical solving of these two challenging problems. In particular, all the existing LSC benchmark instances have been solved thanks to the algorithm presented in [15].…”

“…To sum, the two most recent studies on the PLSE [13] and the LSC [15] significantly contributed to the practical solving of these two challenging problems. In particular, all the existing LSC benchmark instances have been solved thanks to the algorithm presented in [15]. On the contrary, this is not the case for the PLSE and there is still room for improvement in terms of better solving the PLSE instances.…”

“…This work is motivated by this observation, and aims to advance the state-of-the-art of solving the PLSE by establishing new record-breaking lower bounds for the unsolved PLSE instances. For this, we base our work on the proven memetic coloring approach (as demonstrated in, e.g., [10], [15], [22]) and leverage the computing power offered by modern graphical processing units (GPUs). Specifically, we introduce a massively parallel memetic algorithm (MPMA) for the PLSE that evolves in parallel many individuals of a very large population (> 10 4 ) with a high degree of diversity.…”