Solving Graph Coloring Problem by Fuzzy Clustering-Based Genetic Algorithm

Young-Seol Lee and Sung-Bae Cho

Dept. ofComputer Science, Yonsei University, Seoul120-749, Korea tiras@sclab.yonsei.ac.kr , sbcho@cs.yonsei.ac.kr

Abstract. The graph coloring problem is one of famous combinatoria1 optimization problerns. Some researchers attempted to solve combinatoria1 optimization problem with evolutionary a1gorithm, which can find near optima1 solution based on the evolution mechanism of the nature. However, it sometimes requires too much cost to eva1uate fitness of a large number of individuals in the population when applying the GA to the rea1 world problerns. πlÍs paper attempts to solve graph coloring problem using a fuzzy clustering based evolutionary approach to reduce the cost of the eva1uation. In order to show the feasibility of the method, some experiments with other a1ternative methods are conducted.

Keywords: cluster based GA, graph coloring, fuzzy clustering.

1 Introduction

The graph coloring problem is one of the most famous NP-hard problems. The problem requires assignment of colors to each vertex in a given graph with a constraint that two colors assigned to two adjacent vertices must be different. The key point of the problem is to find a minimal number of colors for the color assignment. Some practical applications for optimal resource assignment are related to graph coloring problem. Because of the NP-completeness of the coloring problem, many heuristic methods [1, 2, 3] have been developed.

Genetic a1gorithm is one of the best solutions to solve the graph coloring problem, and it was often used to find optimal solution for many problems such as traveling salesman problem [4] , the quadratic assignment problem [5] and the bin-packing problem [6] and have shown competitive performance. However, genetic a1gorithm requires large population size and much cost to eva1uate the population to discover optimal solution. In some cases with expensive eva1uation cost such as game AI, robot AI and interactive GA, it is difficult to eva1uate a large population in practica1 time. Some researchers have developed a method to estimate fitness values of the whole population by eva1uating the p따t of population [7]. The main point here is to select the P따t of population and reduce eva1uation cost for the graph coloring problem.

In this paper, we present a fuzzy clustering-based genetic algorithm to reduce the cost of coloring problem. It evaluates only the p따t of individuals and estimates fitness

L.T. Bui et al. (Eds.): SEAL 2012, LNCS 7673, pp. 351-360, 2012. @ Springer-Verlag Berlin Heidelberg 2012

352 Y.-S. Lee and S.-B. Cho Solving Graph Coloring P

values of similar individuals with fuzzy integrals. This algorithm allows us to reduce fitness evaluation cost, while maintaining the performance.

item can belong to more th, c1ump algorithm are inc1uded

2 Background 2.3 Related Works

든 든 e낀 ε E , C(Vi) ε C(Vj) U /

1 , ‘ 、

There are many researchers ' lem.Om따i et al. developed 1

heuristic algorithms which a Degree Ordering (SDO) [1]. lem [3]. It avoided cycling al Similarly, Chams et al. appli‘ problem in [1 6]. Brelaz ProF which rely on the comparisOl

developed a heuristic based ! to deal with large scale sche( annealing to solve graph colo rithm using backtracking [18]

Some researchers have stu rumbel et al. presented a hy graph coloring [19]. Eiben et ly changes the fitness functiOl

algorithm combining gene디C tation by tabu search and de nodes (adjacent nodes having resu1ts on the benchmark test! the resu1ts for some large ins evolutionary algorithm and pl ent approaches [21].

2.1 Graph Coloring

Graph coloring is an optimization problem to determine a minimum number (the chromatic number) of color c1asses CJ, C2, ... , Ck in an undirected graph G which has

two components such as nodes V= {v ], ... , V n }, and edges E ={아 | 크 an edge between

Vi and Vj}. The constraint is that Vi and Vj are not in the same color c1ass for each edge e센 든 E [8].

Suppose that C(Vi) be the color (represented by a positive integer) assigned to the node Vi, a proper coloring satisfies the constraint:

In many cases, a random graph, which inc1udes randomly generated edges between nodes with given density d E [0,1], is used to evaluate the performance of an algo-

rithm to s이ve graph coloring. It is difficult to color optimally random graphs having more than 100 nodes [9]. In this paper, we also use random graphs for performance test.

2.2 Clustering Algorithms

Clustering algorithm is to group similar data items automatica11y [10] without any prior knowledge. These groups are named as c1usters. Cluster analysis has been ap­plied in many fields such as image analysis and data mining [11 , 12]. There are three general categories of c1ustering techniques: hierarchical c1ustering, partitional c1uster­ing, and overlapping c1ustering.

Hierarchical c1ustering algorithm constructs hierarchical structures within c1usters. There are two different approaches for hierarchical c1ustering such as bottom-up ap­proach and top-down approach. Bottom-up approach st따ts with n c1usters, and re­peats to merge similar c1usters. Top-down approach divides one c1uster with all data items into some c1usters [10, 13]. There are several hierarchical c1ustering algorithms such as single-linkage algorithm, complete-linkage algorithm, average-linkage algo­rithm, and Ward’s method.

Partitional c1ustering usually generates c1usters that partition the data into similar groups. The goal of the algorithms is to assign data items that are c10se together into a c1uster. In many partitional algorithms, the number of c1usters is determined in ad­vance. K-means c1sutering and hard c-means (HCM) c1ustering are good examples of P따titional c1ustering [10, 13].

Overlapping c1ustering is similar to partitional c1ustering except for no discrete c1usters. In overlapping c1ustering, each c1uster can be overlapped with others and an

3 Proposed Method

The whole algorithm repeats rates initial population and ap uation of them. Only one indi reduce the cost [22]. The indi values of all the population aJ

above process is iterated untl exceeds the fixed number. Tl similar to general evolutionar~

3.1 Encoding and Fitness

There are some encoding sc; encoding as reported in [7].

Solving Graph Coloring Problem by Fuzzy Clustering-Based Genetic Algorithm 353

item can belong to more than one c1uster. Fuzzy c-means (FCM) a1gorithm and b­c1ump algorithm are inc1uded in this category [14, 15].

2.3 Related Works

There are many researchers to develop heuristic methods for a graph coloring prob­lem. Omari et al. developed new heuristic graph coloring algorithms based on known heuristic algorithms which are the Largest Degree Ordering (LDO) and Saturation Degree Ordering (SDO) [1]. Hertz et al. used tabu search to solve the coloring prob­lem [3]. It avoided cyc1ing and local minima using tabu list of forbidden movements. Similarly, Chams et al. applied a simulated annealing algorithm to the graph coloring problem in [16]. Brelaz proposed heuristic methods to color the vertices of a graph which rely on the comparison of the degrees and structure of a graph [2]. Lotfi et al. developed a heuristic based graph coloring algorithm with little computational effort to deal with large scale scheduling problems [17]. Johnson et al. applied a simulated annealing to solve graph coloring problem [9]. Klotz et al. developed a heuristic a1go­rithm using backtracking [18].

Some researchers have studied to find solutions using evolutionary algorithms. Po­rumbel et al. presented a hybrid evolutionary a1gorithm (named as Evocol) for the graph coloring [19]. Eiben et al. used adaptive evolutionary a1gorithm that periodica1-ly changes the fitness function during evolution [20]. Fleurent et al. proposed a hybrid algorithm combining genetic algorithm and tabu search [7]. It improved random mu­tation by tabu search and developed a new crossover operator based on conflicting nodes (adjacent nodes having the same color). The hybrid algorithm has shown good results on the benchmark tests. However, it takes too much computing times to obtain the results for some large instances. Costa et al. compared sequential algorithm and evolutionary algorithm and presented a hybrid algorithm called EDM with two differ­ent approaches [21].

3 Proposed Method

The whole algorithm repeats some operations for evolution process. It frrs t1y gene­rates initia1 population and applies c1ustering technique to the population before eval­uation of them. Only one individual in each c1uster is eva1uated to estimate fitness to reduce the cost [22]. The individual is decided by the centroid in the c1uster. Fitness values of all the population 따e ca1culated from the membership of each c1uster. The above process is iterated until optimal solution is found or the number of iteration exceeds the fixed number. The algorithm is applied to other domains [23, 24]. It is sirnilar to general evolutionary approach except for clustering as shown in Fig. 1.

3.1 Encoding and Fitness Function

There are some encoding schemes such as order-based encoding and string-based encoding as reported in [7]. The string-based encoding is used because it usually

354 Y.-S. Lee and S.-B. Cho

Crossover

•

I\Autation all indivìduafs

General GA Clusteríng based Evaluation

Fig. 1. Fuzzy c1ustering based eva1uation

generates smoother change of fitness than order-based encoding. In the encoding scheme, an individual s for a graph G=(V, E) with n nodes and k the number of avail­able colors is denoted by S=<C(Vl) , C(V2) , ... , C(Vn)> which corresponds to an assign­ment of the k colors to the nodes of the graph. Fig. 2 shows an example of string­based encoding, where Vi denotes the ith node in the graph and 1, 2 and 3 represent first, second, and third colors, respectively. ISI, which is the size of the search space S, is ~ and it can become very large if the graph has more than 100 nodes.

311 1213 11 11 11 1 1 12 I ... [린 Fig. 2. An example of string-based gene encoding with 3 colors

For each individual s, the fitness f(s) can be ca1culated with the number of vi이ated

color constraints as shown in (2).

τ-， llifc(vi )=c(v;) I(s) = 2.. C(l샌) wherec(vi ,v) = • ‘ .J-"l" J/ .. ------"1" J/ 10elsewhere

( Vi꺼 )EE

(2)

where Vi and Vj are the ith and the jth nodes in a given graph and C(Vi) and c(끼) represent the colors of the ith and the jth nodes, respectively. The algorithm airns to reduce 1(s) until f(s)=O for the fixed k to solve a k-coloring problem. 1 point crossover is used and a color of nodes in gene code is changed by mutation.

Solving Graph Coloring Prol

3.2 Fitness Estimation UsÎl

In order to reduce evaluation ‘ evaluate only one individual ir separate the individuals insteac approach is more likely to over‘ because it makes soft boundari ship values [11] .

The fuzzy c-means algorithl proposed by Bezdeck [25]. It much each individual belongs tl between 0 뻐d 1. If the value J

corresponding c1uster. On the c. ciation to the c1uster. It is based

1m = ~

where m is any real number gre c1uster j , X i is the ith of d-dimel the c1uster and 11*11 is any norm and the center. In this case, 1lxt c1uster and an individual Xi ' It

11 xi -c j 11= L> xf :Zth nc

갇 ! 3 1 1 1 2 1

χi [ 3 1 2 1 2 1

찍! o [ 1 I 0 I

Fig. 3. An example of

Fig. 3 shows an example of the parison of colors of nodes betwe

Fuzzy partitioning is carried function shown previously, witl 다 by (5).

Solving Graph Coloring Problem by Fuzzy Clustering-Based Genetic Algorithm 355

3.2 Fitness Estimation Using Fuzzy c-means Clustering

In order to reduce evaluation cost, we separate individuals into several groups and evaluate only one individual in each group. A fuzzy clustering algorithm is used to separate the individuals instead of a hard clustering algorithm. The fuzzy clustering approach is more likely to overcome the local minimum than hard clustering approach because it makes soft boundaries among clusters through the use of fuzzy member­ship values [11].

The fuzzy c-means algorithm is the most widely-used fuzzy clustering algorithm proposed by Bezdeck [25]. It provides fuzzy membership values which mean how much each individual belongs to a specific cluster. The fuzzy membership has a value between 0 and 1. If the value is closer to 0, it indicates a weaker association to the corresponding cluster. On the contr따y， the value closer to 1 indicates a stronger asso­ciation to the cluster. It is based on minimization of the objective function in (3).

Jm=LLU암 11 Xi - C j 112 , 1:::;; m:::;; ∞ (3)

where m is any real number greater than 1, 따 is the degree of membership of Xi in the c1uster j , Xi is the ith of d-dimensional measured data, Cj is the d-dimension center of the c1uster and 11*11 is any norm expressing the similarity between any measured data and the center. In this case, I l.xi-다11 means the distance between centroid Cj of the jth c1uster and an individual Xi. It is calculated as shown in (4).

• j 1 if c(싸) = c(감) 11 X j -c j 11= LC(셔，작) wherec(셔， <)=i

순;‘ 1 0 elsewhere ( 4)

셔 : lth node of X j , <: lth node of C j

갇 ! 3 1 1 1 2 1 3 1 1 1 1 1 1 1 1 1 2 | 단]

χ| 3 1 2 1 2 1 끼 1 1 2 1 1 1 1 1 1 | 단]

펙 o 1 1 1 0 1 끼 0 1 1 1 0 1 0 1 1 I ... [뀐 Fig. 3. An example of distance between an individual Xj and a cluster j

Fig. 3 shows an example of the calculation of the distance which is based on the com­P따ison of colors of nodes between 다 andXi.

Fuzzy partitioning is carried out through an iterative optimization of the objective function shown previously, with the update of membership 따 and the cluster centers 다 by (5).

356 Y.-S. Lee and S.-B. Cho

LU;'Xi

(5)

찮(~: x: : z; :;J꾀 월u암 The iteration stops when ma.xij{ IU/k

+1)_u/k)I}<ë, where ë is a termination criterion be­

tween 0 때d 1, whereas k denotes the iteration step. This procedure converges to a local minimum or a saddle point of Jm .

The fitness values of all individuals are estimated from the fitness of the p따t ofthe individuals and the similarity between the individuals using fuzzy integrals [13] , which are the integrals of a real function with respect to a fuzzy measure.

Let X={x j, X2, ... , Xn} be a set of individuals in the pop버ation 때d C={ C j, C2, ... , cc }

is a set of clusters, and the fitness values of the cluster centers F={/J, /2, ... , .fc}. The fitness values of an individual Xi can be estimated based on mki that means the degree of membership value of the ith individual to the kth cluster center. mki represents the membership of a cluster as a value between 0 때d 1. Because the number of clusters is discrete, the fuzzy integral of k can be calculated by the sum of the values. The esti­mated fitness value of ei is as the following equation:

ei=Lmkix 좌 (6)

4 Experimental Results

4.1 Experimental Settings

A randomly generated graph is used to perform experiments. It is illustrated with a matrix which includes cells with 1 or 0 as shown in Fig. 4. ‘ 1 ’ in the matrix means that the pair of nodes is connected and 0 means not connected. The graph in the expe­riments has 150 nodes and the edges between two nodes are randomly generated with a fixed probability with 0.05.

Table 1 summarizes the parameters of genetic algorithms in the experiments. To compare the performance with altemative methods, various methods summarized in Table 2 are applied to coloring problem in a given random graph. Fuzzy c-means is different from other clustering methods due to including fuzziness of the membership [24].

4.2 Experimental Results

Fig. 5 and 6 show the comparison of fitness changes of an experiment with number of color k = 5. In these figures , x 없is represents the number of generations and y axis

Solving Graph Coloring Pr

N rows

Fig. 4. An example

’fal

Environment

Length of chromosome

Crossover rate

Mutation rate

Max generation

T

Methods FCM Fuzzy c-means clu STD Standard GA with J

KMS K-means clustering SMP Standard GA with I

HCM Hard c-means clust

SLK Single-linkage clus

ALK Average-linkage cl1

CLK Complete-linkage ‘

denotes the number of confli‘ ing problem.

As expected, standard gen ors among various methods. the solution. It requires alm‘ proposed method (FCM) sho' time. Standard genetic algorl problem because it cannot ) such as hard c-means cluster:

Solving Graph Coloring Problem by Fuzzy Clustering-Based Genetic Algorithm 357

fνcolumns

V l 1’, V3 V .(

1' 1 o Q 。

V .J 0 1 o Nrows Vj o o o

V 4 o o o o

Fig. 4. An example of a randomly generated graph for graph coloring

Table 1. Experimental Environment

Environment Value Environment Value

Length of chromosome 150 Fuzziness parameter 1.2

Crossover rate 0.9 Number of clusters 10

Mutation rate 0.005 Number of colors 5 :s n :s 50

Max generation 300

Table 2. Methods in experiments

Methods Description FCM Fuzzy c-means clustering based GA with population size 100 and cluster size 10 STD Standard GA with population size 100 KMS K-means clustering based GA with population size 100 때d cluster size 10

SMP Standard GA with population size 10

HCM Hard c-means clustering based GA with population size 100 and cluster size 10

SLK Single-linkageclustering based(}A with pOjJulation size 100

ALK Average-linkage clustering based GA with population size 100

CLK Compl~!캉1il1kil~(;lustering based GA with population size 100

denotes the number of conflicts. Each line illustrates a method to solve graph color­

ing problem.

As expected, standard genetic algorithm (STD) finds the minimum number of col­

ors among various methods. However, it takes so much evaluation time to discover

the solution. It requires almost ten times more eva1uation than other methods. The

proposed method (FCM) shows the second minimum colors and the second minimum

time. Standard genetic algorithm with sma11 population is not suitable to solve this

problem because it cannot yield competitive solutions. Hard c1ustering techniques

such as hard c-means c1ustering (HCM) and k-means c1ustering (KMS) can generate

358 Y.-S.l,ee and S.-8. Cho

their solution for short evaluation time, but they show worse performance than FCM.The proposed method is a reasonable and competitive solution in some problemswhich require too much evaluation cost. Table 3 summarizes minimum number ofconflicts and average time for three runs.

190

Number

of 170

Conflict150

- . - ALK

1 3 0 r - . F C M

110 - KMS

. -' ' - SLKg A t - . . . . . ,

L 5 1

- H C M

-. - S IMPLE

- . STANDARD

i01 151 241 251

Generetion

Fig. 5. Maximum fitness change with color k=5

Solving Graph Coloring Prc

5 Concluding Remal

This paper proposed an efficieshort time. The method decreauation using fuzzy clustering tals, and evaluates only oneindividuals is indirectly estimaproposed method shows comFthe alternative methods.

There are some problems trof the method is still lower irhave to raise the performance tfuzzy integral including heurisa future work. Also, we willChiarandini's works and confir

Acknowledgments. This reseaCommission), Korea, under thmunications Agency) (KCA-2(

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Keywords: neural rSimplex, evolution, CI\

1 Introduction

Various research studies havwith Artificial Neural Netwicy (controller) of legged rolproven effective lI,2l. Thesrwork architecture via evolul

However, it requires a lawith Neuroevolution algoritrepeating trial-and-error onfore, methods to evolve nerepisodes) are desirable.

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L.T. Bui et al. (Eds.): SEAL 2012@ Springer-Verlag Berlin Heidelbe