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Comparison of the predicted and actual results of the local searches at generation 1, 16 and 31 for the instance DSJC500.5.col.
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Given an undirected graph G=(V,E) with a set of vertices V and a set of edges E, a graph coloring problem involves finding a partition of the vertices into different independent sets. In this paper we present a new framework that combines a deep neural network with the best tools of classical heuristics for graph coloring. The proposed method is ev...
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... 25c.col and le450 25d.col). In Figures 3, and 4, we present two typical patterns of the evolution of the quality of the neural network prediction over the generations that we observed for these instances. For some graphs such as DSJC500.5.col, the neural network makes more and more precise predictions on average over generations, but for other graphs such as wap05a.col, the neural network does not really improve its predictions over time. ...
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... some graphs such as DSJC500.5.col, the neural network makes more and more precise predictions on average over generations, but for other graphs such as wap05a.col, the neural network does not really improve its predictions over time. Figure 3 displays three scatter plots at generation 1, 16 and 31 where, the x-axis and y-axis respectively correspond to the predicted WVCP scores (generation 0, 15 and 30) and the actual WVCP scores (generation 1, 16 and 31) obtained after the local search for the instance DSJC500.5.col for all the p = 20480 individuals of the whole population. In the bottom right corner is displayed a boxplot of the prediction error in percent for the p = 20480 local searches at generation 1, 16 and 31. ...
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... 25c.col and le450 25d.col). In Figures 3, and 4, we present two typical patterns of the evolution of the quality of the neural network prediction over the generations that we observed for these instances. For some graphs such as DSJC500.5.col, the neural network makes more and more precise predictions on average over generations, but for other graphs such as wap05a.col, the neural network does not really improve its predictions over time. ...
Context 4
... some graphs such as DSJC500.5.col, the neural network makes more and more precise predictions on average over generations, but for other graphs such as wap05a.col, the neural network does not really improve its predictions over time. Figure 3 displays three scatter plots at generation 1, 16 and 31 where, the x-axis and y-axis respectively correspond to the predicted WVCP scores (generation 0, 15 and 30) and the actual WVCP scores (generation 1, 16 and 31) obtained after the local search for the instance DSJC500.5.col for all the p = 20480 individuals of the whole population. In the bottom right corner is displayed a boxplot of the prediction error in percent for the p = 20480 local searches at generation 1, 16 and 31. ...
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Citations
... The above hybrid heuristic algorithms are not guaranteed to capture a promising region when restarting the local search from a new point created by the crossover operator. To overcome this issue, Goudet et al (2022). [41] used a deep learning neural network to extract feature information of the structures of graphs, based on which a tabu search strategy was proposed to search for a better solution. ...
... To overcome this issue, Goudet et al (2022). [41] used a deep learning neural network to extract feature information of the structures of graphs, based on which a tabu search strategy was proposed to search for a better solution. However, the algorithm converges slowly, and its performance is limited by hardware since significant computing resources are needed to train the deep learning model. ...
The graph coloring problem (GCP) is a classic combinatorial optimization problem that aims to find the minimum number of colors assigned to vertices of a graph such that no two adjacent vertices receive the same color. GCP has been extensively studied by researchers from various fields, including mathematics, computer science, and biological science. Due to the NP-hard nature, many heuristic algorithms have been proposed to solve GCP. However, existing GCP algorithms focus on either small hard graphs or large-scale sparse graphs (with up to 10^7 vertices). This paper presents an efficient hybrid heuristic algorithm for GCP, named HyColor, which excels in handling large-scale sparse graphs while achieving impressive results on small dense graphs. The efficiency of HyColor comes from the following three aspects: a local decision strategy to improve the lower bound on the chromatic number; a graph-reduction strategy to reduce the working graph; and a k-core and mixed degree-based greedy heuristic for efficiently coloring graphs. HyColor is evaluated against three state-of-the-art GCP algorithms across four benchmarks, comprising three large-scale sparse graph benchmarks and one small dense graph benchmark, totaling 209 instances. The results demonstrate that HyColor consistently outperforms existing heuristic algorithms in both solution accuracy and computational efficiency for the majority of instances. Notably, HyColor achieved the best solutions in 194 instances (over 93%), with 34 of these solutions significantly surpassing those of other algorithms. Furthermore, HyColor successfully determined the chromatic number and achieved optimal coloring in 128 instances.
... Ananda et al. (2022) developed a tabu search graph coloring algorithm for the scheduling problem of nurses in a hospital. Goudet et al. (2022) developed a new framework of deep neural networks with heuristics for the GCP. The algorithm was able to obtain competitive results. ...
Graph Coloring Problem (GCP) is an NP-Hard vertex labeling problem in graphs such that no two adjacent vertices can have the same color. Large instances of GCP cannot be solved in reasonable execution times by exact algorithms. Therefore, soft computing approaches, such as metaheuristics, have proven to be very efficient for solving large instances of GCP. In this study, we propose a new island-parallel ensemble metaheuristic algorithm (PEM-Color) to solve large GCP instances. Ensemble learning is a new machine learning approach based on combining the output of multiple models instead of using a single one. We use Message Passing Interface (MPI) parallel computation libraries to combine recent state-of-the-art metaheuristics: Harris Hawk Optimization (HHO), Artificial Bee Colony (ABC), and Teaching Learning Based (TLBO) to improve the quality of their solutions further. To the best of our knowledge, this is the first study that combines metaheuristics and applies to the GCP using an ensemble approach. We conducted experiments on large graph instances from the well-known DIMACS benchmark using 64 processors and achieved significant improvements in execution times. The experiments also indicate an almost linear speed-up with a strong scalability potential. The solution quality of the instances is promising, as our algorithm outperforms 13 state-of-the-art algorithms.
... The Weighted Vertex Coloring Problem (WVCP), a variant of the GCP, has recently attracted much interest in the literature [3][4][5][6]. In this problem, each vertex of the graph has a weight and the objective is to find a legal coloring such that the sum of the weights of the heaviest vertex of each color group is minimized. ...
... The second category of existing heuristics for the WVCP relies on the population-based memetic framework that combines local search with crossovers. The DLMCOL algorithm [3] of this category uses a deep neural network to learn an invariant by color permutation regression model, useful to select the most promising crossovers at each generation. The AHEAD algorithm [13] involves automatic selection of crossover and local search operators for the WVCP and GCP. ...
... Equation 3 can be compared with the probabilities of choice at each level of the tree by an Ant Colony Optimisation (ACO) algorithm, except that in our case, the choice of the child node is deterministic, and explicitly involves a trade-off between an intensification term, the normalized score of each child node, and an exploration term, depending on its number of visits. ...
This work investigates the Monte Carlo Tree Search (MCTS) method combined with dedicated heuristics for solving the Weighted Vertex Coloring Problem. In addition to the basic MCTS algorithm, we study several MCTS variants where the conventional random simulation is replaced by other simulation strategies including greedy and local search heuristics. We conduct experiments on well-known benchmark instances to assess these combined MCTS variants. We provide empirical evidence to shed light on the advantages and limits of each simulation strategy. This is an extension of the work [1] presented at EvoCOP2022.
... Such a large population has three benefits: (i) it allows to take advantage of massive parallel computation on GPU hardware [13,14]; (ii) it allows to ensure a large diversity of solutions in the population to avoid premature convergence of the algorithm; (iii) it increases the chance for each individual to find a good match in the population for the combination procedure (see Section 3.4). ...
... Replacing the path-linking procedure with the RTSC procedure (columns [11][12][13][14] improves the results for all instances with the same budget of total number of iterations devoted to local search. This improvement is in average of 0.17% for the 17 instances of this table. ...
... GNNs can learn relevant patterns and provide competitive solutions for both vertex coloring and weighted coloring problems. [58] introduced negative message passing in GNNs improves information exchange and accelerates the learning process, leading to better performance in graph coloring tasks [54], [59]. ...
The graph coloring problem functions as a fundamental and pivotal combinatorial optimization task and has played an essential role in various domains such as wireless spectrum management, register planning, and event scheduling. However, traditional coloring algorithms often face limitations such as long computation times and inability to find optimal solutions when dealing with large-scale or complex structured graphs. Against this backdrop, we introduce a graph coloring algorithm underpinned by a Minimal Cost Graph Neural Network (MCGNN). This method incorporates a novel minimum cost optimization mechanism that allows for a deeper exploration of the graph’s structure in comparison to conventional algorithms while leveraging the power of graph neural networks to extract node features for precise graph coloring. Numerical simulations affirm that our scheme not only outperforms existing mainstream methods in finding higher-quality coloring schemes but also does so in reduced computational time.
... This algorithm HEAD is currently one of the most efficient solvers for the k-col. For the WVCP, to our knowledge, there is only one memetic algorithm in the literature, DLMCOL [10], which uses a large population (more than 20,000) and parallel GPU-based local searches, combined with a neural network-guided crossover selection. ...
... The architecture of the AHEAD framework, illustrated in Fig. 1, extends the state-of-the-art HEAD framework [21] by introducing an operator selector. In particular, its simplicity with only two individuals in the population facilitates Author Proof the parent matching and population updating phases compared to other memetic algorithms in the literature [7,10,18,23]. AHEAD takes as input a graph G = (V, E), and an integer value k for the k-col or a weight function w for the WVCP, a set of local search operators O l , a set of crossover operators O x , and a high-level selection strategy π θ characterized by a parameter vector θ. The π θ strategy chooses two pairs of operators <o l , o x > with o l ∈ O l and o x ∈ O x to apply in the current generation to create two new individuals, replacing their parents in the population for the next generation. ...
... This strategy is among the most elitist, as it gives more chances to the best strategy applied in previous iterations. -NN (Neural Network) uses the recommendations of a deep set neural network architecture [10,13,31]. This neural network g θ takes as input a coloring S as a set of k binary vectors v j of size n, S = {v 1 , . . . ...
We present a memetic algorithm with adaptive operator selection for k-coloring and weighted vertex coloring. Our method uses online selection to adaptively determine the couple of crossover and local search operators to apply during the search to improve the efficiency of the algorithm. This leads to better results than without the operator selection and allows us to find a new coloring with 404 colors for C2000.9, one of the largest and densest instances of the classical DIMACS coloring benchmarks. The proposed method also finds three new best solutions for the weighted vertex coloring problem. We investigate the impacts of the different algorithmic variants on both problems.
... Heuristic methods, [30], and approximation algorithms, [20], are used for large instances of graph colouring problems, e.g., genetic algorithm, [2], [10], [12], simulated annealing, [26], local search, [5], memetic algorithm, [8], [13], and also nature-inspired algorithms, e.g., particle swarm optimization, [1], [27], and ant colony optimization, [11]. ...
The graph colouring problem is one of the most studied combinatorial optimisation problems, one with many applications, e.g., in timetabling, resource assignment, team-building problems, network analysis, and cartography. Because of its NP-hardness, the question arises of its solvability for larger instances. Instead of the traditional approaches based on the use of approximate or stochastic heuristic methods, we focus here on the direct use of an integer programming model in the GAMS environment. This environment makes it possible to solve instances much larger than in the past. Neither does it require complex parameter settings or statistical evaluation of the results as in the case of stochastic heuristics because the computational core of software tools, nested in GAMS, is deterministic in nature.
... A variant of the GCP called the Weighted Vertex Coloring Problem (WVCP) has recently attracted much interest in the literature [3][4][5][6]. In this problem, each vertex of the graph has a weight and the objective is to find a legal solution such that the sum of the weights of the heaviest vertex of each color group is minimized. ...
... The second category of existing heuristics for the WVCP relies on the population-based memetic framework that combines local search with crossovers. A recent algorithm [3] of this category uses a deep neural network to learn an invariant by color permutation regression model, useful to select the most promising crossovers at each generation. ...
... Columns 3-8 show the number of instances for which each method can achieve the Best Known Score (BKS) from the literature. Some of the BKS come from [4] (1h runs) or [3] (many hours of calculation), otherwise, they come from our reproductions of the state of the art algorithms. The numbers indicated with a star and in parenthesis in this table correspond to the number of times the method can prove that the best score obtained is optimal. ...
This work investigates the Monte Carlo Tree Search (MCTS) method combined with dedicated heuristics for solving the Weighted Vertex Coloring Problem. In addition to the basic MCTS algorithm, we study several MCTS variants where the conventional random simulation is replaced by other simulation strategies including greedy and local search heuristics. We conduct experiments on well-known benchmark instances to assess these combined MCTS variants. We provide empirical evidence to shed light on the advantages and limits of each simulation strategy. This is an extension of the work of Grelier and al. presented at EvoCOP2022.
... In addition to the exact methods for the WVCP, such as [19,10,5], dedicated heuristics have been proposed in the literature recently, including local search algorithms [20,23,26,28], and memetic algorithms [12]. These heuristics can provide approximate solutions of good quality for medium and large instances, which cannot be solved in a reasonable time by exact methods. ...
... Using the deep set architecture [29,18,12], the output of this neural network is made invariant by the permutation of the color groups in S, so that the neural network selector will take the same decision for two input colors S and S σ which are equivalent up to color group permutation σ. Specifically, for any permutation σ of the input color groups we have ...
... Different highlevel operator selectors have been introduced in this work. The results show that the MCTS versions with adaptive operator selection reach the highest number of best-known scores for the set of 188 benchmark instances of the literature compared to the state-of-the-art methods in one hour of computation time on a CPU (except compared to the DLMCOL algorithm [12] which uses a GPU and was run with an extended computation time). ...
This work presents a hyper-heuristic approach to online learning, which combines Monte Carlo Tree Search with multiple local search operators selected on the fly during the search. The impacts of different operator policies, including proportional bias, one-armed bandit, and neural network, are investigated. Experiments on well-known benchmarks of the Weighted Vertex Coloring Problem are conducted to highlight the advantages and limitations of each dynamic selection strategy.KeywordsMonte Carlo Tree SearchLocal SearchHyper-heuristicWeighted Vertex ColoringLearning-driven optimization
... and dsjr500.5), matching the best-known results [11]. Interestingly, these geometric graphs are not intersection graphs as in the CG:SHOP challenge, but are generated based on a distance threshold. ...
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.
