A Discrete Firefly Algorithm Based on Similarityfor Graph Coloring Problems

Kui ChenDepartment of Computer Science,

Graduate School of Systems and Information Engineering,University of Tsukuba, Japan

Email: chen@kslab.cs.tsukuba.ac.jp

Hitoshi KanohDivision of Information Engineering,

Faculty of Engineering Information and Systems,University of Tsukuba, Japan

Email: kanoh@cs.tsukuba.ac.jp

Abstract—In this paper, we propose a novel non-hybrid discretefirefly algorithm (DFA) for solving planar graph coloring prob-lems. The original FA handles continuous optimization problemsonly. To apply it to discrete problems, we should redefined theoriginal FA over discrete space. In this work, we introduce anew algorithm based on Similarity and discretize FA directlywithout any other hybrid algorithm. The experiments show thatthe proposed method outperforms the success rate of HDPSOand HDABC when solving planar graph coloring problems.

Index Terms—Swarm Intelligence, Graph Coloring Problem,Firefly Algorithm

I. INTRODUCTION

One of the most famous combinatorial optimization prob-lems (COP) is Graph k-coloring problem (GCP). It statesthat given an undirected graph with n nodes and m edges,each node is assigned one of k colors so that two nodesconnected with an edge have different colors. GCP is a kindof NP-complete problems. Because of simple definition, it canbe used to model many scheduling and resource allocationproblems, such as examination scheduling.

Swarm intelligence is the collective hehavior of decentral-ized, self-organized systems. It is inspired from biologicalsystem such as the collective hehavior of birds for foraging anddefending. Swarm algorithms often consist of simple agentswhich not only interact locally with other individuals butalso with the environment. This concept is introduced intocomputer science by Gerardo Beni and Jing Wang [1] and hasbeen successfully applied to many COPs. For example, particleswarm optimization proposed by Kennedy and Eberhart [2],has been extended to solve discrete optimization problems[3] such as graph coloring problem [4], [5], [6], [7] andtimetabling problem [8], [9], [10]. In addition, ant colony opti-mization obtains good results by solving Travelling-SalesmanProblem (TSP) [11], [12] . Finally, the Artificial Bee Colony(ABC) Algorithm proposed by Karaboga and Basturk [13]exhibits excellent results when solving flow shop schedulingproblem [14], [15] and graph coloring problem [16] and [17].

Firefly algorithm (FA) proposed by Xin-She Yang [18] isalso a swarm algorithm. It is inspired by the flashing behaviorof fireflies. Each firefly flashes its light and attracts otherneighbor fireflies. The less brighter one will be attracted by thebrighter one. However, the attractiveness between two fireflies

depends on their distance. That is, the brightness decreases astwo fireflies’ distance increases.

As PSO and ABC, original firefly algorithm is designedto solve continuous optimization problems and only a littleresearch has been done to apply it to the area of combinatorialoptimization problems. In this paper, we propose a discreteFA based on Similarity to solve graph 3-coloring problems.Although hybrid FA for GCP has been proposed [19], fewgeneral-used discrete ABC has been found. In the proposedmethod, the original FA has been divided into two parts anddescritized directly without any hybrid algorithms. Our methodis simple to implement and can be applied to other COPsreadily.

In the following sections, we first introduce the originalFA, GCP and related work. Next, we describe the proposeddescrete firefly algorithm (DFA) in detail. Finally, we comparethe results of experiments solving random generated GCPbetween DFA, HDPSO and DABC and show that our approachcan obtain much higher success rate than the other two.

II. PROBLEM DESCRIPTION

A. Original Firefly Algorithm

In original FA, each firefly represents a solution vector x =[x1, x2, ..., xn] in a given search space and the brightness ofeach firefly can be regarded as the firefly’s fitness. The searchspace is explored by moving firefly xi towards brighter fireflyxj in accordance with below function:

xi = xi + β0e−γr2ij (xj − xi) + α(rand(0, 1)− 1

2) (1)

Note that there are two main parts in the update method, oneis the attractiveness between xi and xj which is determined byβ(r) = β0e

−γr2ij , the other is the random move determinedby the random parameter α. Furthermore, the attractivenessdepends on β0, which is the attractiveness at r = 0, absorbtioncoefficient γ and the Euclidian distance rij between twofireflies.

B. Graph Coloring Problems

Graph coloring problem is an NP-hard problem whichis used to test the efficiency for many newly developedalgorithms frequently. According to Four Color Theorem, the

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solution can be found definitely when 4 colors are used.However, it is more difficult to find a solution when only 3colors are allowed. So, we focus on graph 3-coloring problemsin this paper. We generate random graphs with n nodes andm edges as follows:

1 Dividing nodes into three groups, each with n3 nodes;

2 Adding edges randomly between nodes in differentgroups;

3 Accept the graph until the total number of edges is m.This method is similar but not exactly identical to the methodgiven in [20]. We also define constraint density d as

d =m

n(2)

Constraint density indicates the level of difficulty for a givengraph. Hogg pointed out that the most difficult problems arisewhen d is between 2.0 and 2.5 because the loose constraintsallow for more local optimal solutions [21].

If 0, 1 and 2 represent red(R), green(G) and blue(B)respectively, a candidate solution of GCP can be defined as a1× n vector x = [x1, x2, ..., xi, ..., xn], where xi ∈ {0, 1, 2}.Based on this scheme, one candidate solution of graph in Fig.1can be expressed as [0, 0, 2, 1]:

Fig. 1. An example of graph 3-coloring problem

To evaluate a candidate solution x, we define conflict foreach solution:

conflictjl =

{1 if xj = xl ∧ jl ∈ E(G)

0 otherwise(3)

conflict(x) =1

2

n∑j=1

n∑l=1

conflictjl (4)

where E(G) indicates the set of edges. Eq.3 states that if twoadjacent nodes share the same color, the conflict between thetwo nodes is 1, otherwise 0. The total conflict of solution xis given by Eq.4 obviously. Note that 1

2 is indispensable foravoiding calculating the conflict of two nodes twice.

The objective function used in this work is measured by:

fitness(x) = 1− conflict(x)

m(5)

where x is a candidate solution and m is the edge number ofa given graph. The target of GCP is to find a solution whosetotal conflict is 0, that means the optimal solution is reachedwhen fitness(x) = 1.

C. Related WorkSwarm algorithms have been used to solve many discrete

optimization problems [3]. In this section, we introduce threediscrete swarm algorithms which are proposed to solve graphcoloring problems.

1) Memetic Firefly Algorithm: Memetic firefly algorithm(MFA) is a hybrid algorithm which combined firefly algo-rithm with a heuristical swap local search [19]. Using graph3-coloring problems as test benchmarks, the performanceof MFA was compared with hybrid evolutionary algorithm(HEA), Tabucol and evolutionary algorithm with SAW (EA-SAW). Experiments showed that MFA outperformed otherthree methods when coloring the uniform and equi-partitegraphs but behaved worse when solving flat graph coloringproblems. On the other hand, because it is a hybrid methodand designed for solving graph 3-coloring problems specially,MFA is difficult to be applied to other discrete optimizationproblems.

2) HDPSO: PSO with Transition Probability Based onHamming Distance (HDPSO) is a non-hybrid discrete PSOproposed by Takuya Aoki [7]. In this method, Hammingdistance is used to calculate the distance of two particles inthe swarm. After that, to represent the internal relationshipbetween colors in one particle, three relevant transition proba-bilities are calculated: Prand, Ppbest and Pgbest. Finally, eachcolor in a particle is updated by these three probabilities. Thebasic update method of HDPSO is as follows:

procedure update-position(xi)for j = 1 to N

Generate random number r in [0,1];if r <= Prand

xi[j] = random color;else if r <= Prand + Ppbest

xi[j] = pbest[i][j];else

xi[j] = gbest[j];end if

end forend procedure

Experiments show that HDPSO is an efficient method to solvegraph 3-coloring problem and obtains better results than GA.However, its performance turns to be worse as the size ofgraph becomes large.

3) Discrete ABC: Discrete artificial bee colony (DABC)is also a non-hybrid efficient discrete swarm algorithm [17].It uses Similarity as a criterion of telling distance of twobees. When moving a bee (or a candidate solution) xi, onlysuch xj that is similar to xi can be selected as a neighbor andused to generate a new candidate solution vi. Once a neighborsolution xj is determined, we randomly choose u componentsfrom xj and replace the corresponding components of xi bythem. The procedure of updating is as follows:

procedure update-strategy(swarm)for each solution xi in swarm

vi = xi;randomly select a solution xj;calculate the Similarity of xi and xj;generate a random number r in [0,1];if r < Similarity

randomly choose u componentsfrom xj and replace thecorresponding componentof vi by them;if fitness(vi) > fitness(xi)

replace xi by vi;else

keep xi unchanged;end if

end ifend for

end procedure

Compared with HDPSO, the experiments show that discreteABC obtains higher success rate and lower average evaluation.Especially when applying them to large random graphs, thediscrete ABC keeps relatively high success rate while theperformance of HDPSO becomes worse.

In this paper, we compare our proposed discrete firefly algo-rithm with HDPSO and discrete ABC and test the performanceof them.

III. PROPOSED METHOD

A. Basic Strategy

As section 2.1 shows, the update method of original fireflyalgorithm, which is given by equation (1), has two main parts:(a) β0e−γr

2ij (xj − xi), which moves a firefly xi to another

brighter firefly xj , and (b) α(rand(0, 1) − 12 ), which moves

xi randomly. We call the first part as β-step and the secondpart as α-step. The basic strategy of proposed discrete fireflyalgorithm is simulating the two parts and discretizing themseparately.

B. Discretize β-step with Similarity

In discrete β-step, we use Similarity defined in [7] torepresent the difference between xi and xj :

Similarityij = 1− H(xi,xj)

n(6)

where the H(xi,xj) is the Hamming distance between xi

and xj and n is the dimension of solution vector, which isexactly identical to the node number of a given random graph.Then, we calculate the attractiveness between two fireflies as:

β(xi,xj) = β0e−γSimilarity2ij (7)

After that, we generate a uniformly distributed random numberr from interval [0, 1] for each component k ∈ {1, 2, ..., n} andreplace xik with xjk when r < β(xi,xj). The discrete β-stepcan be shown as below:

procedure discrete beta step (xi, xj)calculate the Similarity;calculate attractiveness beta;for k in [1, 2, ..., n]

generate a random number r;if r <= beta

replace xi[k] with xj[k];

end ifend for

end procedure

C. Discretize α-step

Original α-step adds a small random turbulence to somecomponents of xi. To simulate this procedure in discretespace, α different components of xi are randomly selected andreplaced by other colors. The discrete α-step can be shown asbelow:

procedure discrete alpha step (xi)select alpha different components of xi;replace them by different colors;

end procedure

D. Main Procedure of Discrete Firefly Algorithm

Now, we can build the main procedure of discrete fireflyalgorithm (DFA), which is quite similar with the original one.Let Ii be the brightness of firefly xi and Ii can be calculatedby euqatition (5). In each generation, a firefly is moved towardsanother brighter one based on discrete β-step and α-step. Afterall fireflies’ positions have been updated, we sort firefliesaccording to their brightness and compare the current bestfirefly’s brightness with its parent best firefly’s brightness. Theparent best firefly will take the place of the current best fireflywhen the former is brighter than the latter. The procedure isas follows:

procedure of discrete firefly algorithmInitialization;while (not find optimal solution ||

max iteration is not reached)old_fireflies = new_fireflies;

for (i from 1 to N)for (j from 1 to N)

if (Ii <= Ij)beta-step;alpha-step;

end ifend for

end for

sort new_fireflies;if (current best < parent best)

replace current best with parent best;end if

end whileend procedure

where N is the number of fireflies in swarm.Note that β-step is essentially a monotonically decreasing

function and α-step seems like a random function. If fireflyxi is similar to xj , which means they are in the immediatevicinity, xi is hard to be affected by β-step but only affectedby α-step. On the other hand, if xi is far away from xj , it hasa good changce to be moved towards xj according to β-step.

In short, oiginally near fireflies are avoided becoming closerand closer, and a distant firefly gets a high probability to moveto a better solution. By this strategy, discrete firefly algorithmis getting rid of being trapped in a sub-optimal solution.

IV. EXPERIMENTS

A. Parameter Dependence

Before comparing with HDPSO and DABC, we must exam-ine the dependence of success rate on DFA’s three parameters:α, β0 and γ. Success rate represents the ratio of the times ofsuccessfully finding the optimal solution from all tries. Usingthe method introduced in section 2.2, 30 random graphs with90 nodes (that is n = 90) are generated. The constraint densityd is 2.5, which is the most difficult problem. The swarm size(N ) is 200 and the max iteration is 10000.

Firstly, fixing α on 3 and β0 on 1.0, we test DFA withdifferent γ on the 30 random graphs. The result is shown byFig. 2.

Fig. 2 shows that the best γ is 0.2 with which DFAsuccessfully finding 18 times optimal solution from 30 tries.Accepting 0.2 as the optimal value of γ and fixing α on 3,we test DFA with different β0 to find the optimal value of β0.The result is illustrated in Fig. 3.

We find that the best success rate arises when β0 = 0.9.Finally, fixing γ and β0 on their optimal values, we test DFAwith different α on the 30 random graphs. The result is shownin Fig. 4.

The optimal α is 2 obviously. Using the optimal valuesof parameters of DFA, α = 2, β0 = 0.9, and γ = 0.2, wecompare the proposed method with HDPSO and DABC innext section.

Fig. 2. Relationship between success rate and γ

Fig. 3. Relationship between success rate and β0

Fig. 4. Relationship between success rate and α

B. Comparative Study

The proposed method is compared with HDPSO and DABCon different constraint density d to evaluate its performance.They are evaluated by two measures: success rate and averageevaluation times of objective function. The latter reflects theconvergency speed of an algorithm. The parameters used inHDPSO and DABC are taken from [7] and [17] respectively.For the sake of fairness, 100 independent tries are observed foreach d because of the stochastic nature of swarm algorithms.The experiment conditions are given in Table.I and the resultsare illustrated from Fig. 5 to Fig. 10.

TABLE IEXPERIMENT CONDITIONS

DFA HDPSO DABC

Swarm size 200 200 200Max iteration 10000 10000 10000α 2 – –β0 0.9 – –γ 0.2 – –w – 0.05 –c1 – 7.0 –c2 – 0.03 –limit – – 90u – – 2

Fig. 5. Success rate (n = 90)

Fig. 6. Success rate (n = 120)

Fig. 7. Success rate (n = 150)

As Figs. 5, 6, and 7 show, the proposed DFA obtains anexcellent success rate. When applied to relatively small graphs,say n = 90, the success rate of DFA is higher than HDPSOand nearly equal well with DABC at d = 2.5, which is themost difficult problem. However, as the size of graph increases,DFA brings obvious advantages not only at d = 2.5, but alsoat d = 2 and d = 3. For example, the succcess rate of DFA is78% when the graph size is 150 and d = 2.5. This is muchhigher than DABC’s 20% and HDPSO’s 1%.

The reason is that given a maximum iteration, DFA evalu-ates objective function more times than HDPSO and DABC

do because of its double-loop design. In the comparativeexperiments, we set maximum iteration to 10000 because allthree algorithms can evaluate objective function sufficientlywhen the graph size is 90. If a larger maximum iteration isused, the performance of HDPSO and DABC may improvewhen the graph size is 120 or 150.

Fig. 8. Average evaluation times (n = 90)

Fig. 9. Average evaluation times (n = 120)

Fig. 10. Average evaluation times (n = 150)

On the other hand, as you can see from Figs. 8, 9, and10, though our method performs much better than the othertwo algorithms on the aspect of success rate, its evaluationtimes of solving the most difficult problems are much larger

than HDPSO and DABC. That means DFA spends more timefinding the optimal solution than the other two algorithms. Thisis because there are double loops when updating the position offireflies which denotes the objective function will be calculatedN2 times at the worst case (N is the swarm size). This is aweak point existing in both original FA and proposed DFA.

In summary, the proposed DFA outperforms the successrate of HDPSO and DABC. Especially when the graph size islarge, DFA obtains better success rate compared with other twomethods. However, DFA’s convergency speed is much slowerthan HDPSO and DABC when it is used to solve the mostdifficult problems at d = 2, 2.5, and 3.

V. CONCLUSIONS

In this paper, we proposed a discrete firefly algorithm basedon Similarity and used it to solve graph 3-coloring problems.Experiments on 100 randomly generated graphs show thatthe proposed method obtains good success rate even if thesize of graph is very large. Furthermore, because our methoddiscretizes original firefly algorithm directly wihout any otherhybrid techniques, it can be applied to various large-scale real-life COPs. However, the weak point of the proposed method isalso obvious. It spends more time than its rivals before findingthe optimal solutioin. So, we do not claim that the proposedmethod is the best method to solve graph coloring problems.

In our future work, three things will be done. Firstly,the problem of slow convergency speed will be studied indepth. Second, Hamming distance and Similarity are usedto measure the difference between fireflies in this work. Wewill try other methods of measuring distance and compare theperformance between them. Finally, we will apply this methodto other COPs and test its efficiency.

ACKNOWLEDGMENT

The authors wish to thank Dr. Claus Aranha of University ofTsukuba for his helpful comments and suggestions. This workis supported by JSPS KAKENHI Grant Number 15K00296.

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