Variations on Memetic Algorithms for Graph Coloring Problems

11 pages, 8 figures, 3 tables, 2 algorithmsInternational audienceGraph vertex coloring with a given number of colors is a well-known and much-studied NP-complete problem.The most effective methods to solve this problem are proved to be hybrid algorithms such as memetic algorithms or quantum annealing. Those hybrid algorithms use a powerful local search inside a population-based algorithm.This paper presents a new memetic algorithm based on one of the most effective algorithms: the Hybrid Evolutionary Algorithm HEA from Galinier and Hao (1999).The proposed algorithm, denoted HEAD - for HEA in Duet - works with a population of only two individuals.Moreover, a new way of managing diversity is brought by HEAD.These two main differences greatly improve the results, both in terms of solution quality and computational time.HEAD has produced several good results for the popular DIMACS benchmark graphs, such as 222-colorings for , 81-colorings for and even 47-colorings for and 82-colorings for

Optimality Clue for Graph Coloring Problem

In this paper, we present a new approach which qualifies or not a solution found by a heuristic as a potential optimal solution. Our approach is based on the following observation: for a minimization problem, the number of admissible solutions decreases with the value of the objective function. For the Graph Coloring Problem (GCP), we confirm this observation and present a new way to prove optimality. This proof is based on the counting of the number of different k-colorings and the number of independent sets of a given graph G. Exact solutions counting problems are difficult problems (\#P-complete). However, we show that, using only randomized heuristics, it is possible to define an estimation of the upper bound of the number of k-colorings. This estimate has been calibrated on a large benchmark of graph instances for which the exact number of optimal k-colorings is known. Our approach, called optimality clue, build a sample of k-colorings of a given graph by running many times one randomized heuristic on the same graph instance. We use the evolutionary algorithm HEAD [Moalic et Gondran, 2018], which is one of the most efficient heuristic for GCP. Optimality clue matches with the standard definition of optimality on a wide number of instances of DIMACS and RBCII benchmarks where the optimality is known. Then, we show the clue of optimality for another set of graph instances. Optimality Metaheuristics Near-optimal

Massively parallel hybrid search for the partial Latin square extension problem

The partial Latin square extension problem is to fill as many as possible empty cells of a partially filled Latin square. This problem is a useful model for a wide range of relevant applications in diverse domains. This paper presents the first massively parallel hybrid search algorithm for this computationally challenging problem based on a transformation of the problem to partial graph coloring. The algorithm features the following original elements. Based on a very large population (with more than $10^4$ individuals) and modern graphical processing units, the algorithm performs many local searches in parallel to ensure an intensified exploitation of the search space. It employs a dedicated crossover with a specific parent matching strategy to create a large number of diversified and information-preserving offspring at each generation. Extensive experiments on 1800 benchmark instances show a high competitiveness of the algorithm compared with the current best performing methods. Competitive results are also reported on the related Latin square completion problem. Analyses are performed to shed lights on the understanding of the main algorithmic components. The code of the algorithm will be made publicly available

Using Differential Evolution for the Graph Coloring

Differential evolution was developed for reliable and versatile function optimization. It has also become interesting for other domains because of its ease to use. In this paper, we posed the question of whether differential evolution can also be used by solving of the combinatorial optimization problems, and in particular, for the graph coloring problem. Therefore, a hybrid self-adaptive differential evolution algorithm for graph coloring was proposed that is comparable with the best heuristics for graph coloring today, i.e. Tabucol of Hertz and de Werra and the hybrid evolutionary algorithm of Galinier and Hao. We have focused on the graph 3-coloring. Therefore, the evolutionary algorithm with method SAW of Eiben et al., which achieved excellent results for this kind of graphs, was also incorporated into this study. The extensive experiments show that the differential evolution could become a competitive tool for the solving of graph coloring problem in the future

A Distribution Evolutionary Algorithm for Graph Coloring

Graph Coloring Problem (GCP) is a classic combinatorial optimization problem that has a wide application in theoretical research and engineering. To address complicated GCPs efficiently, a distribution evolutionary algorithm based on population of probability models (DEA-PPM) is proposed. Based on a novel representation of probability model, DEA-PPM employs a Gaussian orthogonal search strategy to explore the probability space, by which global exploration can be realized using a small population. With assistance of local exploitation on a small solution population, DEA-PPM strikes a good balance between exploration and exploitation. Numerical results demonstrate that DEA-PPM performs well on selected complicated GCPs, which contributes to its competitiveness to the state-of-the-art metaheuristics

Conflict Optimization for Binary CSP Applied to Minimum Partition into Plane Subgraphs and Graph Coloring

CG:SHOP is an annual geometric optimization challenge and the 2022 edition proposed the problem of coloring a certain geometric graph defined by line segments. Surprisingly, the top three teams used the same technique, called conflict optimization. This technique has been introduced in the 2021 edition of the challenge, to solve a coordinated motion planning problem. In this paper, we present the technique in the more general framework of binary constraint satisfaction problems (binary CSP). Then, the top three teams describe their different implementations of the same underlying strategy. We evaluate the performance of those implementations to vertex color not only geometric graphs, but also other types of graphs.Comment: To appear at ACM Journal of Experimental Algorithmic

Memetic Artificial Bee Colony Algorithm for Large-Scale Global Optimization

Memetic computation (MC) has emerged recently as a new paradigm of efficient algorithms for solving the hardest optimization problems. On the other hand, artificial bees colony (ABC) algorithms demonstrate good performances when solving continuous and combinatorial optimization problems. This study tries to use these technologies under the same roof. As a result, a memetic ABC (MABC) algorithm has been developed that is hybridized with two local search heuristics: the Nelder-Mead algorithm (NMA) and the random walk with direction exploitation (RWDE). The former is attended more towards exploration, while the latter more towards exploitation of the search space. The stochastic adaptation rule was employed in order to control the balancing between exploration and exploitation. This MABC algorithm was applied to a Special suite on Large Scale Continuous Global Optimization at the 2012 IEEE Congress on Evolutionary Computation. The obtained results the MABC are comparable with the results of DECC-G, DECC-G*, and MLCC.Comment: CONFERENCE: IEEE Congress on Evolutionary Computation, Brisbane, Australia, 201