Graph coloring represents a test bed for many newly developed algorithms because of its simple definition, which states: How to color a graph with the colors, so that none of the vertices connected with an edge have the same color. The coloring is if no two connected vertices are assigned to the same color. A graph is - if it has a proper -. The minimum for which a graph is - is called its .
Many approaches for solving the graph coloring problem (GCP) have been proposed over the time Galinier:2006 ; Malaguti:2009 . The most natural way to solve this problem is, however, in a greedy fashion, where the vertices of the graph are ordered into a permutation, and colored sequential. Thus, the quality of coloring depends on the permutation of the vertices. For example, the DSatur algorithm Brelaz:1979 , one of the best traditional heuristics for graph coloring today, orders the vertices according to . The saturation degree represents the number of distinctly colored vertices adjacent to the vertex . Furthermore, DSatur’s ordering is calculated dynamically during the coloring process.
Many heuristic methods have been developed for larger instances Galinier:2006 because exact algorithms can only color instances of up to 100 vertices. These methods can be divided into local search methods Aarts:1997 and hybrid algorithms Lu:2010 . The most important representative of the former is Tabucol Hertz:1987 , which utilizes the tabu search, as proposed by Glover Glover:1986
. Later were combined local search methods with evolutionary algorithms and improved the results of pure Tabucol, as for example, the hybrid genetic algorithm by Fleurent and FerlandFleurent:1996 , and the hybrid evolutionary algorithm (HEA) by Galinier and Hao Galinier:1999 .
Swarm intelligence is the collective behavior of a self-organized system. Birds, insects, ants, and fish use collective behavior for foraging and defending. These individuals are looking for good food sources and help each other when a lack of food has arisen. This concept was introduced into the computer’s world by Kennedy and Eberhart Kennedy:1995
. Moreover, it was successfully applied to several problem domains, for example, particle swarm optimization, which achieves good results during antenna optimizationRobinson:2004 . In addition, ant colony optimization reaches good results by solving the traveling-salesman person Dorigo:2004 . Finally, the artificial bee colony algorithm, proposed by Karaboga and Basturk Karaboga:2009
, exhibited excellent results when solving combinatorial optimization problemsPan:2011 ; Tasgetiren:2011 .
This paper focuses on the artificial bee colony (ABC) algorithm for graph 3-coloring (3-GCP), which belongs to a class of - Garey:1979 problems. There, the real-valued weights are assigned to the vertices
. These weights determine how difficult the vertex is to color. The higher the weight, the earlier the vertex should be colored. Thus, weights define the order in which the vertices should be colored. This ordering is used by the DSatur traditional algorithm for constructing 3-coloring. The ABC algorithm incorporates DSatur as a decoder. In this manner, the ABC algorithm acts as a meta-heuristic concerned for a generation of new solutions (vector of weights), whilst the quality of the solution (its fitness) is evaluated by DSatur. This approach is not new: it was used by the evolutionary algorithm with SAW method (EA-SAW) of Eiben et al.Eiben:1998 , and by the hybrid self-adaptive differential evolution of Fister et al. Fister:2011 . In the former case, instead of Dsatur, a greedy heuristic was applied as a decoder. Finally, the proposed ABC algorithm was hybridized with a (RWDE) Rao:2009 local search heuristic. This local search heuristic was applied in place of the original sending scouts function and focuses itself on discovering new food sources in the vicinity of the current sources.
The results of the proposed hybrid artificial bee colony algorithm for graph 3-coloring (HABC) was compared with the results obtained with Tabucol, HEA, and EA-SAW for solving an extensive set of random medium-scale graphs generated by the Culberson graph generator Culberson:2011 . A comparison between these algorithms shows that the results of the proposed HABC algorithm are comparable with results of the other algorithms used in the experiments.
The structure of this paper is as follows: In Section 2, the 3-GCP is discussed, in detail. The HABC is described in Section 3, whilst the experiments and results are presented in Section 4. The paper is concluded with a discussion about the quality of the results, and directions for further work are outlined.
Ii Graph 3-coloring
- of a graph is a mapping , where is a set of three colors Bondy:2008 . Note that in the graph definition denotes a set of vertices and a set of edges that associates each edge to an unordered pair of vertices for .
3-GCP can be formally defined as a constraint satisfaction problem (CSP) that is represented as the pair , where with denotes the free search space, in which all solutions are feasible and a Boolean function on (also a feasibility condition) that divides search space into feasible and unfeasible regions. This function is composed of constraints belonging to edges. In fact, to each the corresponding constraint is assigned by if and only if and . Assume that defines the set of constraints belonging to variable . Then, the feasibility condition is expressed as a conjunction of all the constraints .
Typically, constraints are handled indirectly in the sense of the penalty function that transforms the CSP into free optimization problem (FOP) Eiben:2003 (also unconstrained problem). Thus, those infeasible solutions that are far away from a feasible region are punished by higher penalties. The penalty function that is also used as a fitness function here, is expressed as:
where the function is defined as:
In fact, Eq. (1) can be used as a feasibility condition in the sense that if and only if . Note that this equation evaluates the number of constraint violations and determines the quality of solution .
Iii HABC for Graph 3-coloring
In the ABC algorithm, the colony of artificial bees consists of three groups Yang:2008 : employed bees, onlookers, and scouts. The employed bees discover each food source, that is, only one employed bee exists for each food source. The employed bees share information about food sources with onlooker bees, in their hive. Then, the onlooker bees can choose a food sources to forage. Interestingly, those employed bees whose food source is exhausted either by employed or onlooker bees, becomes scouts. The ABC algorithm is formally described in Algorithm 1, from which it can be seen that each cycle of the ABC search process (statements within a while loop) consists of three functions:
SendEmployedBees(): sending the employed bees onto the food sources and evaluating their nectar amounts,
SendOnlookerBees(): sharing the information about food sources with employed bees, selecting the proper food source and evaluating their nectar amounts,
SendScouts(): determining the scout bees and then sending them onto possibly new food sources.
However, before this search process can take place, initialization is performed (function Init()). A termination condition (function TerminationConditionMeet()) is responsible for stoping the search cycle. Typically, the maximum number of function evaluations is used as the termination condition.
The ABC algorithm belongs to population-based algorithms, where the solution of an optimization problem is represented by a food source. The solution of 3-GCP is represented as a real-valued vector for , where denotes the weight associated with the -th vertex of the -th solution; is the number of solutions within the population, and the number of vertices. The values of the weights are taken from the interval , where indicates the lower, and the upper bounds. The initial values of the food sources are generated randomly, according to the equation:
where the function denotes the random value from the interval .
The employed and onlooker bees change their food positions within the search space, according to the equation:
where is a random number from interval
. The onlooker bee selects a food source with regard to the probability value associated with that food sourcecalculated by the equation:
where indicates a mapping from the real-valued search space to the problem search space, as explained in the next subsection, and the fitness function according to Eq. (1).
iii.1 Fitness calculation
The ABC for 3-GCP explores continuous real-valued search space, where the solution is represented as for . Firstly, this solution needs to be transformed into a permutation of vertices . Such a permutation can be decoded into 3-coloring by the DSatur heuristic. The 3-coloring represents the solution of 3-GCP in its original problem space. Whilst a new position regarding a food source is performed within the real-valued search space, its quality is evaluated within the original problem space, according to the equation Eq.(1). This relation can be expressed mathematically as follows:
Note that the function is not injective, i.e. more than one food source can be mapped into the same value of the fitness function. On the other hand, a weakness of this function is that a small move in the real-valued search space can cause a significant increase or decrease in the fitness function.
iii.2 Hybridization with local search
In the classical ABC algorithm, scouts act as a random selection process. That is, if the position of a food source cannot be improved further within a predetermined number of cycles called , then that source is replaced by the randomly generated position. In HABC, instead of randomly generating the new position in the search space (exploration), a deterministic exploitation in the vicinity of the current solution was used Iacca:2011 . Thus, in place of the original SendScouts() function, the RWDE local search heuristic was implemented, which generates the new food sources according to the following equation Rao:2009 :
where is the prescribed scalar step size length and is a unit random vector generated for the -th solution.
Iv Experiments and Results
The goal of the experimental work was to show that HABC can be successfully applied to 3-GCP. In line with this, the proposed HABC was compared with: EA-SAW, Tabucol, and HEA, whose implementations were downloaded from the Internet.
The characteristics of the HABC used during the experiments were as follows: The population size was set at 100 because this value represents a good selection, as was indicated during the experimental work. The value of was set at 1,000, whilst the was limited to 300,000. The former value was obtained through experimental work, whilst the later was selected in order to draw a fair comparison with the other algorithms, i.e. the other algorithms also obey the same limitation. In the end, 25 independent runs were observed, because of the stochastic nature of observed algorithms.
The algorithms were compared according to two measures: (SR) and (AES). The first measure expresses the ratio of the number of successful runs from among all runs, whilst the second reflects the efficiency of a particular algorithm.
iv.1 Test suite
All graphs in the test suite were generated using the Culberson random graph generator Culberson:2011 , which allows to generate graphs of different: size , type , edge probability , and seed . This paper focuses on medium-sized graphs, i.e. graphs with 500 vertices. Three types of graphs were used as follows: uniform (a random graph with variability set at zero), equi-partite, and flat graphs. The edge probability was varied from 0.008 to 0.028 with steps of 0.001. Thus, 21 instances of randomly generated graphs were obtained. Ten different seeds were employed, i.e. from 1 to 10. In summary, each algorithm was solved different instances of graphs.
An interval of edge probabilities was selected such that the region of
was included. Phase transition is a phenomenon that is connected with most combinatorial optimization problems and indicates those regions, where the problem passes over the state of ”solvable” to the state of ”unsolvable”, and vice versaTurner:1988 . The 3-GCP determination of this phenomenon is connected with parameter edge probability. Interestingly, this region is identified differently by many authors. For example, Petford and Welsh Petford:1989 stated that this phenomenon occurs when , Cheeseman et al. Cheeseman:1991 when , and Eiben et al. Eiben:1998 when . In our case, the phase transition needed to be by over Petford and Welsh, by over Cheeseman, and between over Eiben et al..
iv.2 Influence of edge probability
In this experiment, the phenomenon of phase transition was investigated, as illustrated by Fig. 1. The figure is divided into six graphs according to type, and two different measures SR and AES. The graphs capture the results of 21 instances that were obtained by varying the edge probability through a region, including phase transition. Due to space limitation of this paper’s length, a more detailed analysis of the results is left to the reader.
In summary, the best results on medium-sized graphs were reported by HEA and Tabucol. The results of HABC were slightly worse but comparable to both of the mentioned algorithms, whilst the EA-SAW saw the worst results.
iv.3 Influence of the local search
During this experiment, the influence of hybridizing the ABC algorithm with a RWDE local search heuristic was observed. Therefore, a especial focus was placed on the instances during phase transition, i.e. . The two versions of ABC were compared: the original and the hybridized version. In the former, the scouts were generated randomly, whilst in the later the RWDE local search heuristic was used.
The results of this experiment are shown in Table 1, where the row indicates different graph types, whilst the columns and indicate the original and the hybrid ABC algorithms. Note that the average results of the mentioned instances have a varying seed and are presented in the table.
The results showed that using the RWDE local search, substantially improved the results of the original ABC. For example, this improvement amounted to for uniform, for equi-partite, and for flat graphs. On average, hybridization improved the results of the original ABC for .
The results of the proposed HABC for 3-GCP convinced us that the original ABC algorithm is a powerful tool for solving combinatorial optimization problems. HABC gained results that are comparable with the results of the best algorithm for -GCP today (Tabucol and HEA), and improved results obtained with EA-SAW when solving the medium-sized extensive suite of random generated graphs. Note that these graphs are not the hardest to color but are difficult enough that the suitability of the ABC technology for solving the 3-GCP could be successfully proven.
In the future, the HABC for 3-GCP could be additionally improved. In particular, the problem-specific knowledge via local search heuristics could be conducted into the algorithm. The greatest challenge for further work remains the solving of large-scale graph suite (graphs with 1,000 vertices). We are convinced that these graphs could also be successfully solved using the proposed HABC algorithm.
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