Fireworks Algorithm for Graph Coloring

Raouan El Ghazi

, Lamia Benameur

, and Alami Chentoufi Jihane

Faculty of Science, Ibn Tofail University, BP 133, Kenitra, Morocco

Faculty of Science, Abdelmalek Essaadi University, Tetouan, Morocco

Faculty of Science, Ibn Tofail University, BP 133, Kenitra, Morocco

Keywords: Graph coloring problem (GCP), Graph theory, Firework algorithm (FWA), Optimization algorithm, Swarm

intelligence, Metaheuristics.

Abstract: The graph-coloring problem has a long and illustrious history, and it is one of the most famous problems in

the field of graph theory which consists of determining the minimum number necessary to color the vertices

of any graph such that no two adjacent vertices have the same color. In this context, a set of metaheuristics

are proposed to restrict the search for a solution. In this study, a new swarm intelligence optimization

algorithm called fireworks algorithm FWA will be applied to graph coloring problems to speed up the

optimization process, reduce the time, and improve overall performance.

1 INTRODUCTION

The mathematician Leonhard Euler starts coloring

theory specifically in 1736 with the seven bridges of

the Königsberg problem. The problem was a famous

puzzle that concerns the possibility to walk in the

Königsberg by crossing all bridges exactly once.

Then the problem of the four colors always raises the

question of whether four colors are sufficient to draw

any map so that neighboring regions do not have the

same color. Many types of research on the problem

have focused on theoretical aspects such as planar

graphs, triangle-free graphs, and random graphs.

The problem of the coloring of graphs has been

studied by many authors and has developed in various

disciplines like chemistry, biology, social sciences

also in the field of operation research, such as register

allocation (Chaitin;2004), frequency assignment

(Aardel, Hoesel, Koster, Mannino & Sassano;2002),

school and university timetabling (Burke &Newall

;1999) but also for its theoretical aspects. The GCP is

closely associated with a

classical NP-hard

combinatorial optimization problem (Garey &

Johnson; 1979).

Many algorithms have been proposed for this

problem. The DSATUR algorithm (Brélaz, 1979) is

one of the most well-known algorithms due to its

efficiency and simplicity. Leighton proposed another

popular algorithm in the same year. The Recursive

Largest First (RLF) algorithm also the Bruch and-cut

algorithm (Méndez-Diaz & Zabala; 2006). These

methods can only resolve the small size. The

optimization algorithms has become necessary used

for large problems to find an approximate solution in

a reasonable time. Heuristics generally make it

possible to find acceptable solutions in a reasonable

time. However, they do not offer any guarantee as to

the optimality of the best solution found.

For this reason, the researchers are interested in

heuristic approaches. Several metaheuristic methods

had been applied to solve the GCP who seems the

most promising way in finding exciting solutions.

The most efficient heuristic approaches are local

search methods like variable neighborhood search

(Avanthay, Hertz& Zufferey; 2003), tabu search

(Hertz & de Werra; 1987), and variable space search

(Hertz, Plumettaz, and Zufferey; 2008). As well as

hybrid population-based methods such as Genetic and

hybrid algorithms (Fleurent & Ferland, 1996) and

Hybrid evolutionary algorithms (Galinier & Hao,

1999).

Advised Fireworks algorithm has been

increasingly popular in the blind

optimization

algorithms. This algorithm can be used to solve

complex optimization problems. In this paper, we

proposed the new swarm algorithm called fireworks.

FWA is a newly developed in swarm intelligence

optimization algorithm based on the phenomenon of

fireworks explosions in the sky at night. Y. Tan and

Y. Zhu was proposed FWA in 2010 as an

366

El Ghazi, R., Benameur, L. and Chentouﬁ Jihane, A.

Fireworks Algorithm for Graph Coloring.

DOI: 10.5220/0010734500003101

In Proceedings of the 2nd International Conference on Big Data, Modelling and Machine Learning (BML 2021), pages 366-370

ISBN: 978-989-758-559-3

optimization technique. Then it has been successfully

applied in many fields and solving partial

optimization problems, such as multi-satellite control

resource scheduling, hardware/software partitioning,

also, it has resolved image identification and spam

detection, etc....

We organize this paper as follows: In section 2,

we present a mathematical formulation of the graph-

coloring problem. In section 3, an overview of the

fireworks algorithm. Then in section 4, we describe

our proposed method for GCP. In the end, we

conclude and give some perspectives.

2 GRAPH COLORING PROBLEM

Let 𝐺 𝑉,𝐸 be a graph with a set of vertices V

connected by a set of edges E. Given a positive

integer k. A k  coloring of G is a partition of V

into k sets 𝑉1,…,𝑉𝑘 , which called colors

(i.e.,𝑐: 𝑣 1,...,𝑘), the graph coloring problem

seeks to assign each vertex 𝑣∈𝑉 such that no two

vertices in the same set are adjacent.

If two adjacent vertices u and v have the same

color, we say they called conflicting vertices, and the

edge (u; v) ∈ E called a conflicting edge. The k

coloring called legal if there are no conflicting edges

else, it is illegal.

Suppose that f is the objective function that

associates with each coloring the number of

conflicting edges. We can formulate the graph

coloring problem as:

𝑋



𝐺



min





𝑓



𝑥



∶𝑥𝜖𝑆



(1)

𝑋: The set of coloring of the vertices 𝑥𝜖𝑋 can be

uncolored vertices.

𝑆: The set of realizable or legal coloring.

𝑓: Number of colors of the coloring 𝑥𝜖𝑋.

We note 𝐴𝑖,𝑗 the matrix of conflict, and c (i):

the associated color with a vertex 𝑖.



𝑖,𝑗



 

1 if c



𝑖



c



𝑗



𝑎𝑛𝑑



𝑖,𝑗



𝜖𝐸

0 o𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(2)



3 FIREWORKS ALGORITHM

When a fireworks display detonated. A shower of

sparks will fill the local space. This explosion process

is seen as a search in the local space around a specific

point, and when we want to find a point 𝑋



satisfying 𝑓 𝑥



  𝑦, we can continue to set off the

fireworks in the search space until the optimal

solution reached. .For each fireworks display, an

explosion process is initiated and a shower of sparks

fills the local space that surrounds it. The fireworks

as well as the new sparks generated represent

potential solutions in the research space.

We can describe the FWA algorithm as setting off

N fireworks at N different locations, then we will

evaluate the locations of the sparks obtained, the

algorithm ends when the optimal location found.

Otherwise, we will choose another N location and

fireworks for the next generation. Many extensions of

this algorithm saw the light. In 2013, a new method

was presented to calculate the number of explosion

sparks and the magnitude of the explosion of a

firework display (J. Liu, S. Zheng, and Y. Tan, 2013),

and an improved fireworks algorithm was proposed

with five significant improvements. Later in 2014, an

extension of the FWA was presented as the fireworks

algorithm dynamic (Dynamic search in fireworks

algorithm, 2014).

In order to ensure diversity and to balance global

and local research, the explosion magnitude and

population is generated among the new fireworks.

3.1 Explosion

After generating N fireworks randomly, then the

fireworks, N generates sparks in the feasible space.

The explosion operator is a key factor in the algorithm

and plays a very important role. Indeed, the latter

includes the explosion force, the explosion amplitude

and the displacement operation

The number of explosion sparks of each firework is

calculated as follows:

𝑆



𝑚.

















∑





















(3)

Where m is a parameter that controls the overall

number of sparks.

f(x) is an objective function, and x

min

and x

max

denote the limits of the potential space

𝑌



Is the maximum (worst) value of the

objective function among the 𝑛 fireworks.

𝜀 Indicates the smallest constant in the computer,

and it used to prevent the denominator from

becoming zero.

The solution is good (many sparks will be

generated) if the difference between the best and the

bad is within a smaller amplitude. To avoid the

overwhelming effects of fireworks, limits on the

number of sparks 𝑆





are defined, as shown in the

following equation

𝑆





𝑟𝑜𝑢𝑛𝑑



𝑎.𝑚



𝑟𝑜𝑢𝑛𝑑



𝑏.𝑚



𝑟𝑜𝑢𝑛𝑑



𝑆





𝑖𝑓 𝑆



𝑏𝑚 (4)

Fireworks Algorithm for Graph Coloring

367

Where 𝑎 𝑎𝑛𝑑 𝑏 are constant.

After the number of sparks is determined, we will

calculate the amplitude of a particular explosion to

present the best fitness values as follows:

𝐴



𝐴



















∑





















5

𝐴



: Denotes the maximum explosion amplitude.

𝑌



= min f xi i1,2,...,n is the

minimum (best) value of the objective function

among the N fireworks.

ε: has the same meaning.

Then, it is necessary to determine the sparks

displacement in the explosion amplitude.

Displacement operation is to make displacement on

each dimension of a firework. Through the explosion

operator, each firework generates a shower of sparks,

helping to find the global optimal of an optimization

function.

Algorithm 1: Generate a Gaussian spark

1. Calculate the fitness value f (xi) for each

firework.

2. Select randomly z dimensions

z = round (d. rand (0,1))

3. Calculate the coefficient of Gaussian

explosion g = N (1, 1).

4. For each dimension 𝑥





∈{pre-selected z

dimensions of 𝑥





} do

𝑥





𝑥





.𝑔

5. if 𝑥





𝑥





𝑜𝑟 𝑥





𝑥





then

6. 𝑥





𝑥





𝑥





%𝑥





𝑥







7. End if

8. End for

To improve the diversity of the solutions and the

local search ability; the Gaussian sparks strategy is

introduced to produce sparks. The mapping rule will

be carried out to map the spark to a new location

within the feasible space. The sparks of the Gaussian

explosion are shown in Algorithm 1.

3.2 Selection

After generating a shower of sparks, including

explosion sparks and Gaussian sparks. Some of the

generated sparks need to be selected and passed down

to the next generation. In the selection strategy, we

use the measurement of distance. Noted that the best

spark is always kept for the next generation. Then, the

other individuals are selected as:

𝑃



𝑥













∑







∈

(6)

Where 𝑅



𝑥







∑

𝑑𝑥



,𝑥





∑

𝑥



𝑥





∈∈

The distance between 𝑖



solution and all of the other

solutions.

Krepresents all of the solution’s locations.

j∈K means the position j belongs to set K.

Algorithm 2: The pseudo-code of the proposed algorithm

1. Randomly select N fireworks at N

location

2. While terminal criteria = false, do

3. For all firework𝑥



∈ n do

4. Calculate the number and amplitude of

sparks to be generated from the

firework 𝑥



; using Eq. (3), (5).

5. End for

6. For k=1

// 𝒎



is the number of sparks generated by

Gaussian mutation

7. Randomly selects firework that is not in

n set;

8. Explode Gaussian spark (Algorithm 1)

9. End for

10. Select the best sparks from the next

explosion generation;

11. Select the other sparks from both

explosions using based on Eq.6.

12. End While

4 FIREWORKS ALGORITHM

FOR GRAPH COLORING

PROBLEM

For the correspondence with the fireworks algorithm,

we have chosen as the representation of the explosion

space the environment of the graph coloring problems

while the fireworks and the sparks correspond to the

different solutions from the coloration step.

The affinity between a fireworks display and the

sparks generated by the fireworks display measures

the degree of compatibility between them. Affinity is

generally related to distance. However, for the

coloring problem, a cost function is used also

evaluation functions appropriate to the problem. This

function has great importance because it allows

examining if the solution is satisfactory or not.

In general, the cost functions have a combination

of one of the following four components: space,

execution time, communication. In our context, we

consider the following parameters: execution time

and occupied space.

BML 2021 - INTERNATIONAL CONFERENCE ON BIG DATA, MODELLING AND MACHINE LEARNING (BML’21)

368

4.1 The Implementation of the

Fireworks Algorithm

There are several implementations of the fireworks

algorithm. Each one is characterized by its

adaptability to the problem to be solved. In this paper,

we will implement the standard fireworks algorithm.

Several researchers have implemented the

standard fireworks algorithm in several programming

languages (Matlab, C, C ++, JAVA and python).

The algorithm we have proposed is an adaptation

of the standard fireworks algorithm for the graph

coloring problem. The application of this algorithm to

an optimization problem is as follows: Fireworks and

sparks represent the different solutions and the search

space is the environment for graph-coloring problem.

We have proposed the fireworks algorithm to

solve the problem of the coloring of graphs. The

process proceeds as follows: We generate a random

set of N fireworks FW constituting the initial

population (The number of colors equal to k). Then

calculate the cost of each FW using a cost function

(objective function). Select the N best FW with

respect to their cost function. The explosion of the

best FW for a temporary population. Each time a

better solution is found (A valid k-coloring is found)

the population is updated. The Gaussian operation

improves the generated population. Select the best

fireworks from the temporary population then

replaces the N fireworks by the improved FW. The

algorithm stops when the fixed number of iterations

that was carried out and the k-coloring is obtained.

The algorithm that we have proposed requires the

choice of certain parameters, which are as follows:

4.1.1 Generation of the Initial Population

The fireworks algorithm generates an initial

population that represents the set of possible

solutions. The population is generated by the number

of colors k. A vector of integers represents the initial

population whose length is the vertex number.

1 2 3 4 5 6

Figure 1: Generation of the initial population

4.1.2 Objective Function (Fitness)

The Objective function or the adaptation function

assigns each spark to a numerical value. This method

ensures that the best individuals will be kept while the

others are eliminated from the population. In our case,

we try to minimize the number of conflicts (objective

function) by modifying the color of a vertex.

4.1.3 Selection Operator

Selection operator is an operator that consists in

choosing the best sparks and fireworks depends on

the objective function to give birth to a new

generation to apply the processes of explosion and

mutation to them. We have limited our selection to a

random choice of individuals that is done randomly,

uniformly and without the intervention of the fit

value. Each individual therefore has a uniform

probability of being selected.

4.1.4 Mutation Operator

The mutation strategy allows reaching most of the

points of the feasible domain to create new solutions.

We introduce the Gaussian mutation to improve the

diversity of a population. It consists of changing an

invalid coloring 𝛼𝜖1,…,𝑘 for a vertex 𝑖by another

different color. The mutation operator will evolve the

new generation obtained. In the case of binary coding,

it consists of inverting a random bit in a chromosome.

A value bit at 1 takes the value 0 and vice versa.

0 0 1 1 1

1 0 1 1 1

Figure 2: Principle of mutation operation for binary coding

4.1.5 Replacement Operator

After the mutation step, a replacement method is used

to generate a new population so that the size of that

population remains constant. We keep at least the

individual with the best performance from the next

generation. The new generation is made up of the N/2

better and N/2 bad mutated fireworks. While taking

into account the component memory.

After the replacement, some fireworks are

eliminated and replaced by new ones generated

randomly in order to diversify the population. The

lower the affinity of fire, the greater its chance of

being replaced.

The approach used is similar to that of classical

evolutionary algorithms. In the context of

combinatorial optimization problems, we can

consider fireworks algorithms as evolutionary

algorithms presenting particular operators (explosion,

diversification).

Several stopping criteria can be chosen such as the

execution time, the minimum fitness value, the

number of generations. In our study, the chosen

stopping criterion is the number of generations fixed

in advance.

Fireworks Algorithm for Graph Coloring

369

5 CONCLUSIONS

Graph coloring is a classic problem in graph theory

and has attracted the attention of many researchers

because of its multiple practical applications and the

complexity of its resolution. It is difficult to say that

there is a better algorithm for graph coloring; this is

why our choice forced between the simplicity of

execution. In this paper, we have proposed a

Fireworks algorithm to solve the graph coloring

problem. The objective is to have a minimum number

of colors required to color a graph. In our approach,

we have introduced a new feature, although it is

similar to the principles of the algorithm. We found

that the fireworks algorithm is influenced by factors

such as population size, number of iterations.

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