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In this paper a new parallel genetic algorithm for coloring graph vertices is presented. In the algorithm we apply a migration model of parallelism and define two new recombination operators SPPX and CEX. For comparison two problem-oriented crossover operators UISX and GPX are selected. The performance of the algorithm is verified by computer exper...
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Citations
... Its relative pseudocode is as follow [12]. The HB-coloring algorithm uses the crossover greedy partition crossover(GPX), which has been proven experimentally to improve individuals by reducing the number of conflicts [13]. Two mutation operators are used, the first mutation1() is applied to the subset X r as a result of crossing. ...
... Parallel and hybrid metaheuristics are among the most promisssing methods to be developed in the nearest time [2], [20]. Many new algorithms have been already designed and compared with existing methodologies [7], [14], but there is still a room for significant progress in this area. Zbigniew In this article we focus on a class of partitioning problems that appear in many application areas like data clustering [3], column-oriented database partitioning optimization [19], design of digital circuits, decomposition of large digital systems into a number of subsystems (moduls) for multi-chip implementation, task scheduling, timetabling, assiggnment of frequencies in telecommunication networks, etc. Partitionig problems are in general simpler than permutation problems but their search spaces are too huge for exhaustive search or extensive search methods [6], [11], [13], [21], [22]. ...
... Gradually, most graph instances from this repository were assigned these distinctive numbers. The first parallel metaheuristic used for searching chromatic numbers χ(G) was Parallel Evolutionary Algorithm (PEA) [14]. Another example of graph characteristics are graph chromatic sum ∑(G) and graph chromatic sum number s(G) [16]. ...
In
this
article we describe computer experiments while testing a family of parallel and hybrid metaheuristics against a small set of graph partitioning problems like clustering, partitioning into cliques and coloring. In all cases the search space is composed of vertex partitions satisfying specific problem requirements. The solver application contains two sequential and nine parallel/hybrid algorithms developed on the basis of SA and TS metaheuristics. A number of tests are reported and conclusions concerning metaheuristics’ performance that result from the conducted experiments are derived. The article provides a case study in which partitioning numbers \(\psi _{k}(G)\), \(k \ge 2\), of DIMACS graph coloring instances are evaluated experimentally by means of H-SP metaheuristic which is found to be the most efficient in terms of solution quality.
... These graphs are from the well-known DIMACS and COLOR 2002-2004 graph coloring competitions 1 , which are largely tested in the literature for both the vertex coloring and miminum sum coloring problems. The current known "theoretical" bounds of the chromatic sums of the tested instances are shown in Sections 4.1 and 4.2, which are calculated by the number of vertices and edges, as well as the chromatic number χ(G) of each graph [11,13,21,24]. ...
The Minimum Sum Coloring Problem (MSCP) is a relevant model tightly related to the classical vertex coloring problem (VCP). MSCP is known to be NP-hard, thus solving the problem for large graphs is particular challenging. Based on the general “reduce-and-solve” principle and inspired by the work for the VCP, we present an extraction and backward expansion search approach (EBES) to compute the upper and lower bounds for the MSCP on large graphs. The extraction phase reduces the given graph by extracting large collections of pairwise disjoint large independent sets (or color classes). The backward extension phase adds the extracted independent sets to recover the intermediate graphs while optimizing the sum coloring of each intermediate graph. We assess the proposed approach on a set of 35 large benchmark graphs with 450 to 4000 vertices from the DIMACS and COLOR graph coloring competitions. Computational results show that EBES is able to find improved upper bounds for 19 graphs and improved lower bounds for 12 graphs.
... The class of heuristic and metaheuristic algorithms has been mainly developed since 2009 and has enlarged our capacity of finding improved solutions on the benchmark graphs. Representative examples of the existing heuristic algorithms include greedy algorithms (Li et al. 2009;Moukrim et al. 2010), tabu search (Bouziri and Jouini 2010), breakout local search (Benlic and Hao 2012), iterated local search (Helmar and Chiarandini 2011), ant colony (Douiri and Elbernoussi 2012), genetic and memetic algorithms (Douiri and Elbernoussi 2011;Jin and Hao 2016;Jin et al. 2014;Kokosiński and Kwarciany 2007;Moukrim et al. 2013;Wang et al. 2013) as well as heuristics based on independent set extraction Hao 2012, 2013). ...
... They include, for instance, the MASC algorithm (Jin et al. 2014), MA-MSCP algorithm (Moukrim et al. 2013) and the HESA hybrid search algorithm (Jin and Hao 2016). Besides, an early parallel genetic algorithm PGA (Kokosiński and Kwarciany 2007) employs assignment and partition crossovers, first-fit mutation, and proportional selection without any local search improvement. ...
... Recall that for any undirected simple graph G = (V, E) with n = |V | vertices and m = |E| edges, the chromatic number χ(G) is the smallest number of colors needed to color the vertices of G such that a proper k-coloring exists and the chromatic sum (G) is the minimum sum of the colors assigned to all vertices among all proper k-colorings of G. In this section, we list the current known theoretical lower and upper bounds of MSCP according to Kokosiński and Kwarciany (2007), Moukrim et al. (2013) and Thomassen et al. (1989). ...
The Minimum Sum Coloring Problem (MSCP) is a generalization of the well-known
vertex coloring problem. Due to its theoretical and practical relevance, the
MSCP attracts increasing attention. The only existing review on the problem
dates back to 2004 and mainly covers the history of the MSCP and the
theoretical developments on specific graphs. In recent years, the field has
witnessed significant progresses on practical solution algorithms. The purpose
of this review is to provide a comprehensive inspection of the most recent and
representative MSCP algorithms. To be informative, we identify the general
framework followed by these algorithms and the key ingredients that make them
successful. By classifying the main search strategies and putting forward the
critical elements of the reviewed methods, we wish to encourage future
development of more powerful methods and motivate new applications.
... In 2007, Kokosiński and Kwarciany [20] presented the first parallel genetic algorithm which employs assignment and partition crossovers, first-fit mutation, and proportional selection with an island migration model. They showed the first computational results in reference to theoretical upper bounds on 16 small DIMACS graphs. ...
... Otherwise, if P o is no worse than the closest individual P d , then P o replaces P d (lines [16][17]. Otherwise, the worst individual P w is replaced by P o with a small probability of 0.1 (lines [19][20]. This last case corresponds to the situation where P o is close to P d (D min < d α ) and worse than P d (f (P o ) > f (P d )), but P d is different from P w . ...
Given a graph G, a proper k-coloring of G is an assignment of k colors to the vertices of G such that two adjacent vertices receive two different colors. The minimum sum coloring problem (MSCP) is to find a proper k-coloring while minimizing the sum of the colors assigned to the vertices. This paper presents a stochastic hybrid evolutionary search algorithm for computing upper and lower bounds of this NP-hard problem. The proposed algorithm relies on a joint use of two dedicated crossover operators to generate offspring solutions and an iterated double-phase tabu search procedure to improve offspring solutions. A distance-and-quality updating rule is used to maintain a healthy diversity of the population. We show extensive experimental results to demonstrate the effectiveness of the proposed algorithm and provide the first landscape analysis of MSCP to shed light on the behavior of the algorithm.
... The selection of crossover operator has more impact on the performance of GA. The premature convergence [38] in GA can be avoided by selecting appropriate breeding operators. In this paper, the crossover operators are classified in three categories such as standard crossovers, binary crossovers and real/tree crossover s which are application dependant. ...
The performance of Genetic Algorithm (GA) depends on various operators. Crossover operator is one of them. Crossover operators are mainly classified as application dependent crossover operators and application independent crossover operators. Effect of crossover operators in GA is application as well as encoding dependent. This paper will help researchers in selecting appropriate crossover operator for better results. The paper contains description about classical standard crossover operators, binary crossover operators, and application dependant crossover operators. Each crossover operator has its own advantages and disadvantages under various circumstances. This paper reviews the crossover operators proposed and experimented by various researchers.
... Nevertheless, in the last dedicates, metaheuristics are mostly preferred since they provide efficient techniques to explore the search space and then find optimal solutions. For instance, we cite Evolutionary Algorithms [6], Simulated Annealing (SA) [7], Tabu Search Algorithm [8]. ...
... A legal vertex coloring of a graph is a vertex coloring such that two adjacent vertices receive different colors [6]. A proper coloring C using at most k colors is called a proper k-coloring. ...
... For a solution s, the fitness function f (s) is given by equation (6). ...
Flower pollination algorithm is a recent nature inspired algorithm for continuous optimization problems. This algorithm is inspired by the pollination process of flowers. In this paper, we introduce for the first time a hybrid discrete version of the algorithm to deal with the graph coloring problem. The proposed algorithm uses swap and inversion strategies and adapts the efficient constructive method called RLF (Recursive Largest First) to deal with invalid solutions. The performance of the algorithm is evaluated on benchmark instances set and the preliminary results show promising results where it achieves the exact solution in almost cases.
... After a lot of experiments, it proved PGA can improve the efficiency of the sequential genetic algorithm effectively. Some recent work used parallel genetic algorithm [3], [4] which proved to converge more quickly and need less time to obtain the solution than SGA. Therefore, in this paper, we propose a new parallel genetic algorithm for GCP which is based on the CUDA. ...
... The graph coloring problem is to find a partition scheme, so that any two vertices in each subset are not adjacent, therefore, a fitness function ( ) F x is defined to evaluate the graph coloring problem, where x is a chromosome and there are m edges in a graph [3]. ...
Graph coloring problem (GCP) has proven to be an NP hard problem, until now there is no way to solve it in polynomial time. In this paper, a novel parallel genetic algorithm is presented to solve the GCP based on Compute Unified Device Architecture (CUDA). The initialization, crossover, mutation and selection operators are designed parallel in threads. Moreover, the performance of our algorithm is compared with the other graph coloring algorithms using benchmark graphs, and experimental results show that our algorithm converges much more quickly than other algorithms and achieves competitive performance for solving graph coloring problem.
... Nevertheless, in the last dedicates, metaheuristics are mostly preferred since they provide efficient techniques to explore the search space and then find optimal solutions. For instance, we cite Evolutionary Algorithms [6], Simulated Annealing (SA) [7], Tabu Search Algorithm [8]. ...
... A legal vertex coloring of a graph is a vertex coloring such that two adjacent vertices receive different colors [6]. A proper coloring C using at most k colors is called a proper k-coloring. ...
... For a solution s, the fitness function f (s) is given by equation (6). ...
Flower pollination algorithm is a recent nature inspired algorithm for continuous optimization; this algorithm is inspired by the pollination process of flowers. In this paper, we introduce a discrete version of the algorithm to deal with the graph coloring problem. The performance of the algorithm is evaluated on benchmark instances set and the computational results show promising results where it achieves the exact solution in almost cases.
... Considering the theoretical intractability of MSCP, a number of heuristic algorithms have been proposed to find suboptimal solutions, such as a parallel genetic algorithm [10], two greedy heuristics [13], a tabu search metaheuristic [2], a hybrid algorithm (HA) [4], an effective heuristic algorithm (EXSCOL) [21], a local search heuristic (MDS5) [8], and a breakout local search (BLS) [1]. To our knowledge, EXSCOL, HA, MDS5, and BLS are the state-of-theart algorithms in the literature. ...
... MGPX is summarized in Algorithm 2. It builds the color classes of the o offspring one by one, transmitting as many vertices as possible from the parents at each step (for quality purpose) (lines [7][8][9][10][11][12][13][14]. Once a parent has been used for transmitting an entire color class to o, the parent is not considered for a few steps with the purpose of varying the origins of the color classes of o (line 15). ...
... DNTS uses two different and complementary neighborhoods N 1 and N 2 which are applied in a token-ring way [3,15] to find good local optima (intensification) (lines 2-14). More precisely, we start our search with one neighborhood (lines 6-9) and when the search ends with its best local optimum, we switch to the other neighborhood to continue the search while using the last local optimum as the starting point (lines [10][11][12][13]. When this second search terminates, we switch again to the first neighborhood and so on. ...
Given an undirected graph $G$, the Minimum Sum Coloring problem (MSCP) is to
find a legal assignment of colors (represented by natural numbers) to each
vertex of $G$ such that the total sum of the colors assigned to the vertices is
minimized. This paper presents a memetic algorithm for MSCP based on a tabu
search procedure with two neighborhoods and a multi-parent crossover operator.
Experiments on a set of 77 well-known DIMACS and COLOR 2002-2004 benchmark
instances show that the proposed algorithm achieves highly competitive results
in comparison with five state-of-the-art algorithms. In particular, the
proposed algorithm can improve the best known results for 17 instances. We also
provide upper bounds for 18 additional instances for the first time.
