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Variations on Memetic Algorithms for Graph Coloring Problems

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Graph vertices coloring with a given number of colors is a famous and much-studied NP-complete problem. The best methods to solve this problem are hybrid algorithms such as memetic algorithms [Galinier99, Lu10, Wu12] or quantum annealing [Titiloye11a, Titiloye11b, Titiloye12]. Those hybrid algorithms use a powerful local search inside a population-based algorithm. The balance between intensification and diversification is essential for those metaheuristics but difficult to archieve. Customizing metaheuristics takes long time and is one of the main weak points of these approaches. This paper studies the impact of the increase and the decrease of diversification in one of the most effective algorithms known: the Hybrid Evolutionary Algorithm (HEA) from Galinier and Hao [Galinier99]. We then propose a modification of this memetic algorithm in order to work with a population of only two individuals. This new algorithm more effectively manages the correct 'dose' of diversification to add into the included local search - TabuCol [Hertz87] in the case of the HEA. It has produced several good results for the well-known DIMACS benchmark graphs, such as 47-colorings for DSJC500.5, 82-colorings for DSJC1000.5, 222-colorings for DSJC1000.9 and 81-colorings for flat1000\_76\_0, which have so far only been produced by quantum annealing [Titiloye12] in 2012 with massive multi-CPUs.
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Variations on Memetic Algorithms
for Graph Coloring Problems
Laurent Moalic1and Alexandre Gondran2
1IRTES-SET, UTBM, University of Technology of Belfort-Montbliard, Belfort, France
2MAIAA, ENAC, French Civil Aviation University, Toulouse, France
Abstract
Graph vertices coloring with a given number of colors is a famous and much-
studied NP-complete problem. The best methods to solve this problem are hybrid
algorithms such as memetic algorithms [15, 26, 38] or quantum annealing [34, 35,
36]. Those hybrid algorithms use a powerful local search inside a population-based
algorithm. The balance between intensification and diversification is essential for
those metaheuristics but difficult to archieve. Customizing metaheuristics takes
long time and is one of the main weak points of these approaches. This paper
studies the impact of the increase and the decrease of diversification in one of
the most effective algorithms known: the Hybrid Evolutionary Algorithm (HEA)
from Galinier and Hao [15]. We then propose a modification of this memetic
algorithm in order to work with a population of only two individuals. This new
algorithm more effectively manages the correct ‘dose’ of diversification to add
into the included local search - TabuCol [20] in the case of the HEA. It has
produced several good results for the well-known DIMACS benchmark graphs,
such as 47-colorings for DSJC500.5, 82-colorings for DSJC1000.5, 222-colorings
for DSJC1000.9 and 81-colorings for flat1000 76 0, which have so far only been
produced by quantum annealing [36] in 2012 with massive multi-CPUs.
1 Introduction
Given a undirected graph G= (V, E) with Va set vertices and Ea set of edges, the
graph vertex coloring involves assigning each vertex with a color so that two adjacent
vertices (linked by an edge) feature different colors. The Graph Vertex Coloring Prob-
lem (GVCP) involves finding the minimum number of colors required to color a given
graph in respect of these binary constraints. The GVCP is a famous and much-studied
problem because this simple formalization can be applied to various issues such as
laurent.moalic@utbm.fr
alexandre.gondran@enac.fr
1
arXiv:1401.2184v1 [cs.AI] 8 Jan 2014
frequency assignment problems [1, 10], scheduling problems [25, 39, 37] and fly level
allocation problems [3]. Most problems that involve sharing a rare resource (colors) be-
tween different operators (vertices) can be modeled as a GVCP, such as most resource
allocation problems. GVCP is NP-hard [18]. Given ka positive integer corresponding
to the maximum number of colors, a k-coloring of a given graph Gis a function cthat
assigns to each vertex a color (i.e. an integer included between 1 and k) as follows :
c:V→ {1,2..., k}
v7→ c(v)
One recalls some definitions : a k-coloring is called legal or proper k-coloring if it
respects the following binary constraints : (u, v)E, c(u)6=c(v). Otherwise the k-
coloring is called non legal or non proper; and edges (u, v)Esuch as c(u) = c(v) are
called conflicting edges, and uand vconflicting vertices. A given graph Gis k-colorable
if a proper k-coloring exists. The chromatic number χ(G) of a given graph Gis the
smallest integer ksuch as Gis k-colorable. A coloring is called complete coloring if one
color is assigned to all vertices, otherwise it is called partial coloring, whereby some
vertices may remain uncolored. An independent set or stable set is a set of vertices, no
two of which are adjacent. In this case, it is possible to assign the same color to all the
vertices of an independent set without producing any conflicting edges. The problem of
finding a minimal graph partition of independent sets is then equivalent to the GVCP.
The k-coloring problem - finding a proper k-coloring of a given graph G- is NP-
complete [24] for k > 2. Therefore, the best performing exact algorithms are generally
not able to find a proper k-coloring in reasonable time when the number of vertices is
greater than 100 [22, 11]. For large graphs, one uses heuristics that partially explore
the search space to occasionally find a proper k-coloring in a reasonable time frame.
However, this partial search does not guarantee that a better solution exists. Very
interesting and comprehensive surveys on the GVCP and the most effective heuristics
to solve it can be found in [16, 14]. These studies classify heuristics by the search space
used. In order to define the search space (or that which is termed strategy), one has to
answer to three questions :
1. Is the number of available colors fixed or not ?
2. Is non proper colorings included in the search space ?
3. Does the heuristic use complete colorings or partial colorings ?
Among the eight theoretical possibilities, four main strategies are defined [14] :
Proper strategy if the number of colors is not fixed and only complete and proper
colorings are taken into account. The aim is to find a coloring that minimizes
the number of colors used under constraints of legality and completeness of the
coloring.
k-fixed partial proper strategy if the number of colors is fixed and partial and
proper colorings are taken into account. The aim is to find a coloring that mini-
mizes the number of uncolored vertices under constraints of the number of given
colors and of proper coloring.
2
k-fixed penalty strategy if the number of colors is fixed and complete and no proper
colorings are taken into account. The aim is to find a coloring that minimizes the
number of conflicting edges under constraints of the number of given colors and
of completed coloring.
Penalty strategy if the number of colors is not fixed and no proper and completed
colorings are taken into account. The aim is to find a coloring that minimizes the
number of conflicting edges and the number of colors used under the constraint
of completed coloring.
The Variable Space Search of [21] is interesting and didactic because it works with three
of the four above strategies. Another more classical means of classifying the different
methods is to consider how these methods explore the search space.
Constructive or exhaustive methods (such as greedy methods, branch and bound,
backtracking and constraint programming) build a coloring step-by-step from
empty coloring; those approaches usually have a k-fixed partial proper strategy.
DSATUR [5] and RLF [25] are the most well-known greedy algorithms. Those
algorithms rapidly provide an upper bound on χ(G), but it is quite distance from
the exact value of χ(G). They are used to initialize solutions before a local search
or an evolutionary algorithm is employed. Some improved algorithm such as
XRFL [22] based on RLF provides much better results. Exact algorithms such as
branch and bound (B&B), backtracking, or constraint programming [6] are k-fixed
partial proper strategies. The B&B implementation of [22] takes too much time
after 100 vertices. Column generation approaches are also exact methods. [27]
divided the problem into two parts: the first problem involves generating useful
columns in the form of independent sets. The second problem involves selecting
a minimum number of independent sets (created by the first problem) in order to
cover the graph. To obtain better results, exact methods can be hybridized with
local searches [32, 9].
Local (or neighborhood or trajectory) searches (such as the hillclimbing, the simu-
lated annealing [22], the tabu search [20, 8], the variable neighborhood search [2],
the variable space search [21] and the min-conflict) start from an initial coloring
and try to improve it by local moves; usually those approaches have a proper
or k-fixed penalty strategies. A detailed overview of those local search methods
for graph coloring is provided in [16]. In our algorithm, we employ an improved
version [15] of a tabu search algorithm called TabuCol [20]; it is one of the first
metaheuristics developed for graph coloring and uses a k-fixed penalty strategy.
Population-based approaches (such as the evolutionary algorithm, the ant colony
optimization [29], the particle swarm algorithm and the quantum annealing [34,
35]) work with several colorings that can interact together with, for example, a
crossover operator, a share memory, a repulsion or attraction operator, or others
criteria such as sharing. Those methods currently obtain the best results when
they are combined with the aforementioned local search algorithms. However,
used alone, those population-based approaches are limited.
3
The first objective of this paper is to present a new and simple algorithm that pro-
vides the best results for coloring DIMACS benchmark graphs. The second objective
is to show why this algorithm effectively controls diversification. Indeed, in our work,
the population of this memetic algorithm is reduced to only two individuals, called a
couple-population. It provides the opportunity to focus on the correct ‘dose’ of diversity
to add to a local search. This new approach is simpler that a general evolutionary algo-
rithm because it does not use numerous parameters such as selection, size of population,
crossover and mutation rates.
The organization of this paper is as follows. The issue of diversification in heuristics
is presented in Section 2. Section 3 describes our improvement of the memetic algorithm
for the GVCP. The experimental results are presented in Section 4 and some of the
impacts of diversification are analyzed in Section 5. Finally, we consider the conclusions
of this study and discuss possible future research in Section 6.
2 How to manage diversity ?
Because the search is not exhaustive, heuristics provide only sub-optimal solutions with
no information about the distance between those sub-solutions and the optimal solution.
Heuristics return the best solution found after partial exploration of the search space. To
produce a good solution, heuristics must alternate exploitation phases and exploration
phases. During an exploitation phase (or an intensification phase), a solution produced
by the heuristic is improved. It resembles an exhaustive search but occurs in a small
part of the search space. An exploration phase (or diversification phase) involves a
search into numerous uncharted regions of the search space. The balance between
those two phases is difficult to achieve. Metaheuristics therefore define a framework
that manages this balance. In this section, we classify the main components of several
metaheuristics as intensification operators or as diversification operators. Of course,
this list, presented in table 1, is not exhaustive and we take into account only well-
known metaheuristics : Local Search (LS), Tabu Search (TS), Simulated Annealing
(SA), Variable Neighborhood Search (VNS), Evolutionary Algorithm (EA), and Ant
Colony (AC). Some components can be shared between several metaheuristics. Other
components can be at the same time an intensification operator and a diversification
operator, such as parent selection or population update. It depends on the context of
the algorithm, as we will now demonstrate.
Local search algorithms start from an initial solution and then try to improve this
iteratively through local transformation. The simplest LS, hill-climbing, accepts a move
only if the objective function is improved: it is inherently an intensification operator.
It is possible to introduce some diversity by generating several different initial solu-
tions. This process is called multi-starting. The limit of a simple LS is that after a
given number of iterations, the algorithm is blocked within a local optimum. No local
transformation can improve the solution. SA and TS therefore accept some worsening
moves during the search, with a given criteria for SA and with a tabu list for TS. On
the other hand, VNS, to escape local optimum, changes the neighborhood structure.
The change in neighborhood structure is an effective diversification operator because it
4
never worsens the current solution. However, this diversification operator is too weak:
VNS must add a shaking process, similar to partial restart (as in Iterated Local Search),
in order to achieve a more global optimization.
The population-based algorithms are often classified as global optimization algo-
rithms because they work with several candidate solutions, although this does not indi-
cate that the algorithm will find the global optimum. Moreover, working with several
candidate solutions can be regarded as a diversification operator. EAs are population-
based algorithms. A basic EA can be presented in five steps as follows: 1) Parents
selection: one selects two individuals of the population, which are called parents. 2)
Crossover: according to a given rate, a crossover operator is applied to those two par-
ents, which creates two new individuals, called children, blending of the two parents.
3) Mutation: according to a given rate, a mutation operator is applied to each children,
which modifies them slightly. 4) Population update: under given conditions, the two
children take the place of two individuals of the population. 5) This cycle of four steps
is called a generation; it is repeated until a given stop condition is realized. The best
individual of the population is the output of the EA. We shall classify the different
components of this basic EA as intensification operators or diversification operators.
Mutation and crossover operators are different in nature to selection and population
update processes. Indeed, mutation and crossover operators are chance generators,
while selection and population update processes are higher-level systems that control
the chance interest. These two means of functioning are pithily summarized in an
expression attributed to Democritus: “Everything existing in the universe is the fruit
of chance and necessity”, which is also considered in Jacques Monod’s book Chance and
Necessity [28]. The mutation and the crossover processes are therefore diversification
operators in essence. The mutation operator slightly changes one of the individuals
of the population. The crossover operator mixes two individuals of the population in
order to create a new one. The mutation is a unary operator while the crossover is a
binary operator. There exists also trinary operators for some other EAs, such as for
Differential Evolution. The unary or binary changes (applied by mutation or crossover)
are performed randomly and these are therefore exploration phases. Occasionally, a
random modification improves the solution, but this function is driven by the higher-
level system. These changes cannot be considered intensification operators, except in
some specific cases where the modifications (mutation or crossover) are not performed
randomly. For example, in [12], the crossover is a quasi-deterministic process directly
guided by the separability of the objective function in order to improve the latter.
The aim of a diversification operator is to provide diversity to current solution(s), but
there is a risk of providing too much diversity, and thus breaking the structure of current
solution(s). The crossover operator generally provides more diversity than the mutation
operator.
The classification of operators such as the parents selection and the population
update depends on the choice of the EA parameters. Indeed, if one chooses an elitist
parents selection policy (the choice of best individuals of the population for the crossover
and the mutation), the selection is then an intensification operator. Conversely, a policy
of random parents selection (a random selection of individuals from the population
for the crossover and the mutation) indicates a diversification operator. This double
5
Intensification/Exploitation Diversification/Exploration
LS: accept improving moves LS: multi-starts
SA, TS: accept worsening moves
VNS: change of neighborhood structure VNS: change of neighborhood structure
shaking
p-bA: population (several candidate solutions)
EA: parents selection (the best) EA: parents selection (random)
population update (if child is better) population update (systematic)
mutation, crossover
sharing, elitism
AC: collective memory (pheromone) AC: random choice
Table 1: Classification of main components of some well-known metaheuristics as inten-
sification operators or as diversification operators. LS: Local Search, TS: Tabu Search,
SA: Simulated Annealing, VNS: Variable Neighborhood Search, EA: Evolutionary Al-
gorithm, AC: Ant Colony, p-bA: population based Algorithm.
role of the parents selection can be very interesting but also very difficult to properly
control. The population update has the same feature. If one chooses to include in
the population the individuals created by the crossover and/or the mutation (children)
only if they are better than the individuals of the population that one removes, then
the population update is an intensification operator. Conversely, if one chooses to
systematically replace some individuals of the population by the created children (even
if they are worse), then the population update is a diversification operator. The sharing
and the elitism are two others mechanisms of EAs playing diversification operators roles.
In AC, the interactions between individuals of the population occur through the
sharing of a collective memory. The phenomenon mechanism (deposit and evaporation)
plays this role, while the random choice of the instantiation of variables plays the role
of diversification.
Table 1 summarizes the classification of main components of several metaheuristics
as diversification operators or as intensification operators. The separation is not always
very clear, such as in the case of EAs, in which parents selection or update population
processes play both roles. Controlling the correct balance between intensification and
diversification is a difficult step; Much fine tuning is required to obtain good results
from heuristics.
An interesting feature of a diversification operator is its ability to explore new areas
of the search space without breaking the structure of current solution(s). We recall a
quotation of the physiologist Claude Bernard about hormones [4] and that corresponds
well with the role of a diversification operator :
“Everything is poisonous, nothing is poisonous, it is all a matter of dose.”
Claude Bernard - 1872
The level or dose of diversity provided by a diversification operator is difficult to eval-
uate. However, it is possible to compare the diversification operators with their dose of
diversity. For example, the mutation operator generally provides less diversity than the
crossover operator. For each operator, there are many parameters that affect the dose
6
Figure 1: Fitness of the child solution in function of the hamming distance separating
the two parents having the same fitness. The fitness is the number of conflicting edges
in a completed 48-coloring. The fitness of the parents is equal to f= 40. The hamming
distance separating two k-colorings is the minimum number of moves (changing of one
vertex of color class) to pass from one k-coloring to the other (it is independent of the
colors symmetry).
of diversity; One of the easily identifiable parameters for the crossover operator is the
Hamming distance between the two parents. Indeed, if we highlight two good solutions
that have very good objective functions but that are very different (in terms of Ham-
ming distance), then they would have a high probability of producing children with bad
objective functions following crossover. The danger of the crossover is of completely
destroying the solution structure. On the other hand, two solutions very closed (in
terms of Hamming distance) produce a child with quasi-similar fitness. Chart 1 shows
the correlation between the Hamming distance separating the two parents of the same
fitness (abscissa axis) and the fitness of the child (ordinate axis). This chart is obtained
for k-coloring problem where the objective is to minimize the number of conflicting
edges (the fitness) in a complete but non legal k-coloring with k= 48. We consider the
DSJC500.5 graph from the DIMACS benchmark [23] and the GPX crossover of [15].
The parents used for the crossover have the same fitness value, equal to 40 conflicting
edges. There is a quasi-linear correlation between these two parameters.
The key question is : how does one manage the correct ‘dose’ of diversification in a
heuristic? In the next section, we present a memetic algorithm with only 2 individuals,
which simplifies the management of the diversity dose.
7
3 Memetic Algorithms with only two individuals
Memetic Algorithms [19] (MA) are hybrid metaheuristics using a local search algorithm
inside a population-based algorithm. They can also be viewed as specific EAs where
all individuals of the population are local minimum (of a specific neighborhood). This
hybrid metaheuristic is a low-level relay hybrid (LRH) in the Talbi taxonomy [33].
A low-level hybridization means that a section of one metaheuristic is replaced by a
second metaheuristic. In MA, the mutation of the EA is replaced by a local search
algorithm. Conversely, a high-level hybridization refers to a combination of several
metaheuristics, none of which are modified. A relay (respectively teamwork ) hybrid
means that metaheuristics are used successively (respectively in parallel).
In graph coloring, the Hybrid Evolutionary Algorithm (HEA) of Galinier and Hao [15]
is a MA; the mutation of the EA is replaced by a tabu search. HEA is one of the best
algorithms for solving the GVCP; From 1999 until 2012, it provided the majority of
the best results for DIMACS benchmark graphs [23], particularly for difficult graphs
such as DSJC500.5 and DSJC1000.5 (see table 2). These results were obtained with a
population of 8 individuals.
The tabu search used is an improvement of the TabuCol of [20]. This version of
TabuCol is a very powerful tabu search, which even obtained very good results for the
GVCP when it is used alone (see table 2). Another benefit of this version of TabuCol is
that it has only two parameters to adjust in order to control the tabu tenure. Moreover,
[15] have demonstrated on a very large number of instances that with the same setting,
TabuCol obtained very good k-colorings. Indeed, one of the main disadvantages of
heuristics is that the number of parameters to set is high and difficult to adjust. This
version of TabuCol is very robust. Thus we retain the setting of [15] in all our tests
and consider TabuCol as an atomic algorithm.
The mutation, which is a diversification operator, is replaced by a local search,
an intensification operator. The balance between intensification and diversification is
restored because in the case of the graph coloring problem, all crossovers are too strong
and destroy too much of a solution’s structure.
The crossover used in HEA is called the Greedy Partition Crossover (GPX); it is
based on color classes. Its aim is to add slightly more diversification to the TabuCol
algorithm.
These hybridizations combine the benefits of population-based methods, which are
better at diversification by means of a crossover operator, and local search methods,
which are better at intensification.
The intensification/diversification balance is difficult to achieve. In order to simplify
the numerous parameters involved in EAs, we have chosen to consider a population with
only two individuals. This implies 1) no selection process, and 2) a simpler management
of the diversification because there is only one diversification operator: the crossover.
8
3.1 General Pattern - H2O: Hybrid approach with 2 trajectories-
based Optimization
The basic blocks of HEA are the TabuCol algorithm, which is a very powerful local
search for intensification, and the Greedy Partion Crossover (GPX), which adds a little
more diversification.
The idea is to combine as simply as possible these two blocks. The local search is a
unary operator while the crossover is a binary operator. We present with algorithm 1
the pseudo code of a first version of this simple algorithm, which can be seen as a
2 trajectories-based algorithm. We then call the algorithm ‘H2O’, which signifies a
Hybrid approach with 2 trajectories-based Optimization.
Algorithm 1 First version of H2O: Hybrid approach with 2 trajectories-based Opti-
mization
1: Input: an asymmetric crossover, a local search.
2: Output: the best solution found best
3: p1, p2, best init() {initialize with random solutions}
4: generation 0
5: repeat
6: c0
1crossover(p1,p2)
7: c0
2crossover(p2,p1)
8: c1localSearch(c0
1)
9: c2localSearch(c0
2)
10: best saveBest(c1, c2, best)
11: p1, p2replace(c1, c2)
12: generation + +
13: until stop condition()
14: return best
The algorithm 1 needs an asymmetric crossover, which means: crossover(p1,p2)6=
crossover(p2,p1). After randomly initializing the two solutions, the algorithm repeats
an instructions loop until a stop criteria occurres. First, we introduce some diversity
with the crossover operator, then the two offspring c0
1and c0
2are improved by means of
the local search. Next, we register the best solution and we systematically replace the
parents by the two children. An iteration of this algorithm is called a generation.
In order to add more diversity on the algorithm 1, we present a second version of
H2Owith two extra elite solutions.
9
Algorithm 2 Second version of H2O: Hybrid approach with 2 trajectories-based
Optimization
1: Input: an asymmetric crossover, a local search, Itercycle .
2: Output: the best solution found
3: p1, p2, elite1, elite2, best init() {initialize with random solutions}
4: generation 0
5: repeat
6: c0
1crossover(p1,p2)
7: c0
2crossover(p2,p1)
8: c1localSearch(c0
1)
9: c2localSearch(c0
2)
10: elite1saveBest(c1, c2, elite1)
11: p1, p2replace(c1, c2)
12: best saveBest(elite1, best)
13: if generation%Itercy cle = 0 then
14: p1elite2
15: elite2elite1
16: elite1init()
17: end if
18: generation + +
19: until stop condition()
20: return best
We add two other candidate solutions (similar to elite solutions), elite1and elite2,
in order to reintroduce some diversity in the couple-population. Indeed, after a given
number of generations, the two individuals of the population become increasingly similar
within the search space. To maintain a given diversity in the couple-population, the
elite2elite solution replaces one of the population individual after I tercycle generations
i.e. one cycle. elite1is the best solution found during the current cycle and elite2the
best solution found during the previous cycle. The figure 2 represents the graphic view
of the algorithm 2.
3.2 Application to graph coloring : H2col
H2col is the application of H2Oto the k-coloring problem with the GPX as crossover
and the TabuCol as local search. It uses only one parameter: IterT C , the number of
iterations performed by the TabuCol algorithm.
10
Crossover'
Local'search'
P1# P2#
cycle'itera/ons'
best1#
?#
?#
best2#
Figure 2: Diagram of H2O.
Algorithm 3 H2col : Hybrid approach with 2 trajectories-based Optimization for
graph coloring with GPX and TabuCol
1: Input: the asymmetric crossover: GPX, the local search: TabuCol (with IterT C
iterations), Itercycle = 10.
2: Output: the best configuration found
3: p1, p2, elite1, elite2init() {initialize with random colorings}
4: generation 0
5: repeat
6: c0
1GPX(p1,p2)
7: c0
2GPX(p2,p1)
8: c1TabuCol(c0
1)
9: c2TabuCol(c0
2)
10: elite1saveBest(c1,c2,elite1)
11: p1c1
12: p2c2
13: best saveBest(elite1, best)
14: if generation%Itercy cle = 0 then
15: p1elite2
16: elite2elite1
17: elite1init()
18: end if
19: generation + +
20: until nbConf licts > 0
21: return best
11
This algorithm is an improvement of the TabuCol algorithm; these are two parallel
TabuCol algorithms that interact periodically by crossover. We will now briefly recall
the principles of the TabuCol algorithm and the GPX crossover.
3.2.1 TabuCol
In 1987, [20] presented the TabuCol algorithm, one year after Fred Glover introduced
the tabu search. This algorithm, which solves k-coloring problems, was enhanced in
1999 by [15]. The three basic features of this trajectory-based algorithm are as follows:
Search Space and Objective Function: the algorithm is a k-fixed penalty strategy.
The objective function minimizes the number of conflicting edges.
Neighborhood: ak-coloring solution is a neighbor of an other k-coloring solution
if the color of only one conflicting vertex is different. This move is called a critic
1-move. Therefore the neighborhood size depends on the number of conflicting
vertices.
Move Strategy: the move strategy is the standard tabu search strategy. Even if
the objective function is worse, at each iteration, one decides to move to the best
neighbors, which are not inside the tabu list. Note that all the neighborhood is
explored. If there are several best moves, one chooses one of them at random.
This is the only random aspect of this metaheuristic. The tabu list is not the
list of each already-visited solution because this is computationally expensive. It
is more efficient to put only the reverse moves inside the tabu list. Indeed, the
aim is to forbid returning to previous solutions, and it is possible to reach this
goal by forbidding the reverse moves during a given number of iterations (i.e.
the tabu tenure). The tabu tenure is dynamic: it depends on the neighborhood
size. A basic aspiration criteria is also implemented: it accepts a tabu move
to a k-coloring, which has a better objective function than the best k-coloring
encountered so far.
Data structures have a major impact on algorithm efficiency, constituting one of the
main differences between the Hertz and de Werra version of TabuCol [20] and the
Galinier and Hao version [15]. Checking that a 1-move is tabu or not and updating the
tabu list are operations in constant time (figure 3b). TabuCol also uses an incremental
evaluation [13]: the objective function of the neighbors is not computed from scratch,
but only the difference between the two solutions is computed. This is a very important
feature for local search efficiency. Finding the best 1-move corresponds to finding the
maximum value of a matrix (cf. figure 3a).
3.2.2 Greedy Partition Crossover (GPX)
The second block of H2col is the Greedy Partition Crossover (GPX) from the Hybrid
Evolutionary Algorithm HEA [15]. The two main principles of GPX are: 1) a coloring
is a partition of vertices and not an assignment of colors to vertices, and 2) large color
classes should be transmitted to the child. The figure 4 gives an example of GPX for
12
(a) Matrix of incremental evaluation. It gives
the benefit of the all 1-moves; for example,
changing the color of the C vertex in green adds
2 conflicts; the cross, for example in (A,blue),
means that the A vertex is colored in blue in
the current solution; therefore it is not a real
1-move.
(b) Matrix representing the tabu list and
the tabu tenure. If the current iteration
is the 21st iteration, therefore all the 1-
moves of a value greater than 21 means
that this 1-move is tabu; for example,
(B,blue)-move is tabu because 34 >21.
Figure 3: Data structures for TabuCol algorithm.
a problem with three colors (red, blue and green) and 10 vertices (A, B, C, D, E, F,
G, H, I and J). The first step is to transmit to the child the largest color class of the
first parent. After having withdrawn those vertices in the second parent, one proceeds
to step 2 where one transmits to the child the largest color class of the second parent.
This process is repeated until all the colors are used. There are most probably still
some uncolored vertices in the child solution. The final step is to randomly add those
vertices to the color classes.
4 Experimental Results
In this section we present the results obtained with the two versions of the proposed
memetic algorithm; the first version of H2col without elites solutions is indicated as
H0
2col, and the second version with the two extra elites solutions is simply indicated
H2col. Test instances are selected among the most studied graphs since the 1990s, which
are known to be very difficult (DIMACS [23]). To validate the proposed approach, the
results of H2col are compared with the results obtained by some of the best methods
currently known.
4.1 Instances and Benchmarks
We study some graphs from the second DIMACS challenge of 1992-1993 [23]. This is
to date the most widely-used benchmark for solving the graph coloring problem. These
instances are available at the following address: ftp://dimacs.rutgers.edu
We focus on two main types of graphs from the DIMACS benchmark: DSJC and
FLAT, which are randomly or quasi-randomly generated. DSJCn.d graphs are graphs
with nvertices, wich each vertex connected to an average of n×dvertices; dis the
13
Figure 4: An example of GPX crossover for a graph of 10 vertices (A, B, C, D, E, F,
G, H, I and J) and three colors (red, blue and green). This example comes from [15].
graph density. The chromatic number of these graphs is unknown. FLAT graphs have
another structure: they are built for a known chromatic number. The flatn χ graph
has nvertices and χis the chromatic number.
4.2 Computational Results
H2col and H0
2col were programmed in C++ standard. The results presented in this
section were obtained on a computer with an Intel Xeon 3.10GHz processor - 4 cores
and 8GB of RAM. Note that the RAM size has no impact on the calculations: even
for large graphs such as DSJC1000.9 (with 1000 vertices and high density of 0.9), the
memory used does not exceed 125 MB. The main characteristic is the processor speed.
As shown in Section 3, the proposed algorithms have two successive calls to local
search (lines 8 and 9 of the algorithms 1, 2 and 3), one for each child of the current gen-
eration. Almost all of the time is spent performing the local search. It is possible and
moreover easy to parallelize both local searches when using multi-core processor archi-
tecture. This is what we have done using the OpenMP API (Open Multi-Processing),
which has the advantage of being cross-platform (Linux, Windows, MacOS, etc.) and
simple to use. The execution times provided in the following table are in CPU time.
Thus, when we give an execution time of 30 minutes, the required time is actually close
to 15 minutes using two processing cores.
Table 2 presents results of the principal methods known to date. For each graph, it
indicates the lowest number of colors found by each algorithm. For TabuCol [20] the
reported results are from [21] which are better than those of 1987. The most recent al-
gorithm, QA-col (Quantum Annealing for graph coloring [36]), provides the best results
14
Graphs H2COL
LS Hybrid algorithm
2013 1987/2008 1999 2008 2010 2011 nov. 2012
H2col TabuCol [20, 21] HEA [15] AmaCol [17] MACOL[26] EXTRACOL [38] QA-col [36]
DSJC500.1 12 13 - 12 12 - -
DSJC500.5 47 50 48 48 48 - 47
DSJC500.9 126 127 - 126 126 -126
DSJC1000.1 20 -20 20 20 20 20
DSJC1000.5 82 89 83 84 83 83 82
DSJC1000.9 222 227 224 224 222 222 222
flat1000 50 0 50 50 -50 50 50 -
flat1000 60 0 60 60 -60 60 60 -
flat1000 76 0 81 88 83 84 82 82 81
Table 2: Best coloring found
but is based on a cluster of PC using 10 processing cores simultaneously. Note that
HEA [15], AmaCol [17], MACOL [26] and EXTRACOL [38] are also population-based
algorithms also using TabuCol and GPX crossover or an improvement of GPX (GPX
with n>2 parents for MACOL and EXTRACOL and the GPX process is replaced in
AmaCol by a selection of kcolor classes among a very large pool of color classes). Only
QA-col has another approach based on several parallel simulated annealing algorithms
interacting together with sharing criteria.
In a standard Simulating Annealing algorithm (SA), the probability of accepting
a candidate solution is managed through a temperature criteria. The value of the
temperature decreases during the SA iterations. A Quantum Annealing (QA) is a
memetic algorithm without crossover and in which the local search is a SA. The only
interaction between the individuals of the population occurs through a specific sharing
process. A standard sharing process involves penalizing solutions that are too ‘close’
within the search space (the simplest sharing is to forbid having two similar solutions
in the population). However, comparing a solution with all solutions of the population
can be computationally expensive, especially for a large population. This is why QA-col
compares a solution with only two others of the population: this comparison provides
the solution-population distance. The value of this measure is integrated into the
temperature value of each SA. If the solution-population distance is greater, then the
temperature will be higher, and there will be a higher probability that the solution will
be accepted. [30, 31] have also developed a specific population spacing management
that resembles this one. However, there are different ways to calculate the distance
between two solutions. For a set of solutions, [7] define frozen same pairs (respectively
frozen different pairs) as pairs of vertices that are in the same color class (respectively
in the different color class) for all solutions. In QA-col, authors use these definitions
with a set of two solutions in order to define a distance between these two solutions
as the difference between the number of frozen same pairs and the number of frozen
different pairs.
Table 3 presents the results obtained with H0
2col, the first version of H2col (without
elites). This simplest version finds very good solutions (+1 color compared to the
best known results) for difficult graphs within the literature. Only one method, QA-
col, occasionally finds a solution with less color. The column IterTC indicates the
number of iterations of the TabuCol algorithm (this is the stop criteria of TabuCol).
The column Success evaluates the robustness of this method, providing the success
rate: success runs/total runs. A success run is that which finds a legal k-coloring. The
15
Instances k IterTC Success Iter Gene Time
DSJC500.1 12 8000 15/20 2.5×106158 0.2 min
DSJC500.5 48 8000 9/20 5 ×106356 0.5 min
DSJC500.9 126 25000 10/20 15 ×106317 2 min
DSJC1000.1 20 7000 6/20 6 ×106472 0.9 min
DSJC1000.5 83 40000 16/20 137 ×1061723 36 min
DSJC1000.9 223 30000 4/20 56 ×106939 12 min
222 60000 1/20 516 ×1064304 131 min
flat1000 50 0 50 130000 20/20 1.1×1065 0.5 min
flat1000 60 0 60 130000 20/20 2.4×1069 1 min
flat1000 76 0 82 40000 19/20 152 ×1061905 38 min
81 30000 1/20 380 ×1066333 118 min
Table 3: Results of H0
2col, the first version of H2col algorithm (without elites)
average number of generations or crossovers performed during one success run is given
by Gene value. The total average number of iterations of TabuCol preformed during
H0
2Ois Iter =IterTC ×Gene ×2. The column Time indicates the average CPU time
in minutes of success runs.
H0
2col does not find the solutions each time for these graphs, but when it does, it is
generally very rapid. For example, for the graph coloring DSJC1000.1 with 20 colors,
recent results reported in the literature are:
MACOL in 108 minutes (CPU 3.4GHz) [26]
EXTRACOL in 93 minutes (CPU 2.8GHz) [38]
Our algorithm, H0
2col, achieves solutions in less than one minute (CPU 3.1GHz).
The main drawback of H0
2col is that it converges sometimes too quickly. In such
instances it cannot find a solution before the two individuals in a generation become
identical. The second version, H2Col, adds more diversity while performing an intensi-
fying role.
Table 4 shows the results obtained with H2Col. Of primary important is that
H2Col finds solutions with fewer colors than all the best-known methods. Only the
Quantum Annealing algorithm, using ten CPU cores simultaneously, achieves this level
of performance. In particular, DSJC500.5 is solved with only 47 colors and flat1000 76 0
with 81 colors. We noted 1* the number of success runs when the solution is occasionally
found, but on average this occurs in less than one in 20 runs. This is the case for graphs
DSJC500.5 and flat1000 76 0 with 47 colors and 81 respectively.
The computation time of H2Col is generally close to that of H0
2Col but the former
algorithm is more robust with quasi-100% of success. In particular, the two graphs
DSJC500.5 and DSJC1000.1 with 48 and 20 colors respectively are resolved each time,
and in less than one CPU minute on average. Using a multicore CPU, these instances
are solved in less than 30 seconds on average, often in less than 10 seconds. As a
comparison, the shortest time reported in the literature for DSJC1000.1 is 93 minutes
16
Instances k IterTC Success Iter Gene Time
DSJC500.1 12 4000 20/20 3.8×106483 0.3 min
DSJC500.5 48 8000 20/20 7 ×106494 0.9 min
47 12000 1* 20 ×106850 3 min
DSJC500.9 126 15000 13/20 29 ×106970 4 min
DSJC1000.1 20 3000 20/20 4 ×106346 0.7 min
DSJC1000.5 83 40000 20/20 96 ×1061200 16 min
82 40000 2/20 548 ×1066854 96 min
DSJC1000.9 223 30000 19/20 126 ×1062107 32 min
222 50000 1/20 1.4×10914208 354 min
flat1000 50 0 50 130000 20/20 1.1×1065 0.5 min
flat1000 60 0 60 130000 20/20 2.2×1069 1 min
flat1000 76 0 82 40000 20/20 84 ×1061052 16 min
81 40000 2/20 716 ×1068961 116 min
Table 4: Results of the second version of H2col algorithm (with elites) with the indica-
tion of CPU time
for EXTRACOL with a 2.8GHz processor (and 108 minutes for MACOL with a 3.4GHz
processor).
5 Analysis of diversification
In this section we perform several tests in order to analyze the role of diversification
in the H2col algorithm. We increase or decrease the dose of diversification within the
H2col algorithm. There are only two operators that lead to diversification: the GPX
crossover and the population update process. In a first set of tests, we slightly modify
the dose of diversification in the GPX crossover and analyze the results. In a second
set of tests, we focus on the population update process: in H2col, the two produced
children systematically replace both parents, even if they have worse fitness values than
their parents. If we lighten this rule, the diversification decreases.
5.1 Dose of diversification in the GPX crossover
Some modifications are performed on the GPX crossover in order to increase (as for
the first test) or decrease (as for the second test) the dose of diversification within this
operator.
5.1.1 Test on GPX with increased chance: random draw of a color classes
number
In order to increase the levels of chance within the GPX crossover, one randomizes
the GPX. One recalls (cf. section 3.2.2) that at each step of the GPX, the selected
parent transmits the largest color class to the child. In this test, we begin by randomly
17
transmitting xcolor classes chosen from the parents to the child; after those xsteps,
one starts again by alternately transmitting the largest color class from each parent. x
is the random level. If x= 0, then the crossover is the same as the initial GPX. If x
increases, then the chance and the diversity also increase. To evaluate this modification
of the crossover, we count the cumulative iterations number of TabuCol that one H2col
run requires in order to find a legal k-coloring. For each xvalue, the algorithm runs
ten times in order to produce more robust results. For the test, we consider the 48-
coloring problem for graph DSJC500.5 of the DIMACS benchmark. The figure 5 shows
in abscissa the random level xand in ordinate the average necessary iterations number
required to find a legal 48-coloring.
Figure 5: Average necessary iterations number to find a legal 48-coloring for DSJC500.5
graph in function of the randomness level; abscissa: x, the randomness level; ordinate:
average iterations number
First, 0 xk, with kthe number of colors, but we stop the computation for
x15, because with x= 15, the algorithm does not find a 48-coloring within acceptable
computing time limit. This means that when we introduce too much diversification,
the algorithm cannot find a legal solution. Indeed, for xhigh value, the crossover does
not transmit the good features of the parents, so that the child appears to be a random
initial solution. For 0 x8, the algorithm finds a legal coloring in more or less 10
million iterations. It is not easy to decide which x-value obtains the quickest result.
5.1.2 Test on GPX with decreased chance: imbalanced crossover and im-
portant different between the two parents
In the standard GPX, the role of each parent is balanced: they alternatively transmit
their largest color class to the child. Of course, the parent, which first transmits its
largest class, has more importance than the other; this is why it is an asymmetric
18
crossover. In this test, we give a higher importance to one of the parents. At each
step, we randomly draw the parent that transmits its largest color class with a different
probability for each parent. We introduce x, the probability of selecting the first parent;
1xis the probability of selecting the second parent. For example, if x= 0.75, parent
1 always has a 3 in 4 chance of being selected to transmit its largest color class (parent
2 only has a 1 in 4 chance). If x= 0.5, it means that both parents have an equal
probability (a fifty-fifty chance) to be chosen; this almost corresponds to the standard
GPX. If x= 1, it means that the child is a copy of parent 1; there are no more crossovers
and therefore H2col is a TabuCol with two initial solutions. When xget away from
0.5, the chance and the diversity bring by the crossover decrease. Figure 6 shows in
abscissa the probability xand in ordinate the average number of necessary iterations
required to find a legal 48-coloring (as in the previous test).
Figure 6: Average number of necessary iterations required to find a legal 48-coloring
for DSJC500.5 graph in function of the imbalanced crossover; abscissa: x, probability
to select the first parent; ordinate: average iterations number
First, it is evident that the results are symmetric according to x. The best results
are obtained with approximately x= 0.5 (0.4x0.6). The impact of this parameter
is weaker than that of the previous one: the control of the reduction of diversification
is finer.
5.2 Test on parents’ replacement: systematic or not
In H2col, the two produced children systematically replace both parents, even if they
have worse fitness values than their parents. We modify this replacement rule in this
test. If the fitness value of the child is better than that of its parents, the child automati-
cally replaces one of the parents. Otherwise, we introduce a probability xcorresponding
to the probability of the parents’ replacement, even if the child is worse than his par-
ents. If x= 1, the replacement is systematic as in standard H2col and if x= 0, the
19
replacement is performed only if the children are better. When the x-value decreases,
the diversity also decreases. Figure 7 shows in abscissa the parents’ replacement prob-
ability xand in ordinate the average number of necessary iterations required to find a
legal 48-coloring (as in the previous test).
Figure 7: Average number of necessary iterations required to find a legal 48-coloring
for DSJC500.5 graph in function of the parents’ replacement policy; abscissa: parents’
replacement probability; ordinate: average number of iterations
If the parents’ replacement probability x= 0 or a very low 0.1, then more time is
required to produce the results. The absence or the lack of diversification is shown
to penalize the search. However, for a large range of values: 0.3x1, it is not
possible to define the best policy for xcriteria. The dramatic change in behavior of
H2col happens very quickly around 0.2.
These studies enable us to better understand the role of the diversification operators
(crossover and parent updating). Some interesting criteria are identified as the random
level of the crossover or the imbalanced level of the crossover. We will integrate these
criteria that we have studied in this paper into future algorithms in order to dynamically
manage the diversity.
6 Conclusion
We have proposed a new algorithm for the graph coloring problem, called H2col. It is
a variation of a memetic algorithm with only two candidate solutions. This simplifi-
cation has the great advantage of clarifying the role of the diversification and intensi-
fication operators and of more effectively managing the right ‘dose’ of diversification.
H2col combines a local search algorithm (TabuCol) as an intensification operator with
a crossover operator (GPX) as a diversification operator, two main building blocks of
Galinier and Hao’s memetic algorithm [15]. The computational experiments, carried
20
out on a set of challenging DIMACS graphs, show that H2col finds the best existing
results, such as 47-colorings for DSJC500.5, 82-colorings for DSJC1000.5, 222-colorings
for DSJC1000.9 and 81-colorings for flat1000 76 0, which have so far only been found
by quantum annealing [36] with a massive multi-CPU.
We have performed an in-depth analysis on the crossover operator in order to bet-
ter understand its role in the diversification process. Some interesting criteria have
been identified, such as the crossover’s levels of randomness and imbalance. Those
criteria pave the way for our further research. We have generalized this optimization
methodology, which is a specific memetic algorithm with only two individuals in its
population. This Hybrid approach with 2 trajectories-based Optimization (H2O) im-
proves the local search algorithm through a crossover operator. The crossover inserts a
dose of diversification that is then easy to manage.
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... Nevertheless, these studies rarely exploit specific knowledge of the problem, which makes these learning approaches more general but may limit their performance. Indeed, the results obtained by this type of approach are for the moment far from the results obtained by state of the art algorithms on graph coloring problems such as hybrid algorithms [27,29,31] and simulated annealing algorithms [44]. We can mention however recent studies which try to take advantage of efficient local search algorithms and machine learning techniques [14,49] with promising results for graph coloring problems. ...
... We do not consider performing crossovers between individuals too far away in the search space as this may result in poor quality offsprings (cf. [31]). ...
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... In 2015, Moalic and Alexandre (Moalic and Gondran, 2015) proposed a hybrid evolutionary algorithm based on two individuals (HEAD), which reduced computation time. In 2017, Moalic and Alexandre (Moalic and Gondran, 2017)introduced random crossover and unbalanced crossover in HEAD to improve the performance of the algorithm. Goudet and Duval (Goudet et al., 2020)et al. proposed a population-based weight learning framework to solve the GCP. ...
... Hybrid evolutionary algorithms are a class of algorithms that combine local search algorithms with evolutionary algorithms and are often used to solve optimization problems. In 2017, Moalic et al. gave a hybrid evolutionary algorithm HEAD (Moalic and Gondran, 2017), as in Algorithm 3. HEAD removes the complex selection and update operators from the evolutionary algorithm and uses an elite strategy to manage population diversity. ...
... Moalic and Alexandre proposed two versions of HEAD (Moalic and Gondran, 2017), the first version without the strategy of adding elite individuals, which allows Algorithm 4 :NERS HEAD; // HEAD with new elite replacement strategy Input: K is the number of colors, Iter T C is the number of Tabucol iterations, generation is the number of evolutive generations, Maxgeneration is the maximum number of evolutive generations, ϕ = 5, ε = 0.1; Output: the gbest scheme s; 1: P = {P 1 , P 2 , elite 1 , elite 2 , temp, gbest} ← init(); 2: generation ← 0;f pre ← 999;f curr ← 0; 3: while f (gbest) > 0 and P 1 = P 2 and generation < Maxgeneration do 4: ...
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The graph coloring problem is an NP-hard problem. Currently, the most effective method to solve the problem is the hybrid algorithm. This paper proposes a hybrid evolutionary algorithm NERS_HEAD with a new elite replacement strategy. In NERS_HEAD, a method to detect the local optimal state is proposed so that the evolutionary process can jump out of the local optimal by introducing diversity on time. A new elite structure and replacement strategy are designed to increase the diversity of the evolutionary population so that the evolution process can not only converge quickly but also jump out of the local optimum in time. Experiments based on 34 instances of the DIMACS benchmark show that compared with the current excellent graph coloring algorithm, NERS_HEAD can reduce the evolutionary generation by an average of 28.3% and the calculation time by 22.11%. Especially in the instance dsjc500.1, Nere_HEAD can reduce 56% of evolution generations and 40% of computing time; On the r1000.1c instance, it can reduce the evolution generations by 82.30% and the calculation time by 73.45%.
... Inspired by [27,17], FTMS maintains a small population of only three individuals, whose rationale is explained in Section 4.3. The overall framework of FTMS is shown in Algorithm 1. FTMS consists of four main modules: population initialization, inherit-repair recombination (IRR), betweenness centrality-based late-acceptance search (BCLS) and population updating. ...
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The distance-based critical node problem involves identifying a subset of nodes in a graph such that the removal of these nodes leads to a residual graph with the minimum distance-based connectivity. Due to its NP-hard nature, solving this problem using exact algorithms has proved to be highly challenging. Moreover, existing heuristic algorithms are typically time-consuming. In this work, we introduce a fast tri-individual memetic search approach to solve the problem. The proposed approach maintains a small population of only three individuals during the whole search. At each generation, it sequentially executes an inherit-repair recombination operator to generate a promising offspring solution, a fast betweenness centrality-based late-acceptance search to find high-quality local optima, and a simple population updating strategy to maintain a healthy population. Extensive experiments on both real-world and synthetic benchmarks show our method significantly outperforms state-of-the-art algorithms. In particular, it can steadily find the known optimal solutions for all 22 real-world instances with known optima in only one minute, and new upper bounds on the remaining 22 large real-world instances. For 54 synthetic instances, it finds new upper bounds on 36 instances, and matches the previous best-known upper bounds on 15 other instances in ten minutes. Finally, we investigate the usefulness of each key algorithmic ingredient.
... Moalic et al. proposed a hybrid evolutionary algorithm (HEAD) based on meme strategy to solve the graph coloring problem and solve the DIMACS benchmark graph. The simulation experiment shows that the algorithm achieves good results compared with the comparison algorithm [57]. ...
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The classic combinatorial optimization problem of graph coloring is one of the most famous NP-complete problems. One example of the graph coloring problem is the four-color map problem. There have been many applications of swarm intelligence optimization algorithms to this problem, but to date, such algorithms can only solve the four-color map problem with fewer than 100 regions. This article proposes an enhanced discrete dragonfly algorithm (EDDA) for four-color map problems. We use global and local discrete alternate search strategies-when there is at least one adjacent dragonfly around the i-th dragonfly, a global search is performed; when there are no other dragonflies around, a local search is performed. A greedy strategy, local differential cross strategy, and single-point switching strategy are then used to solve the problem of conflicts among adjacent nodes. Finally, six real-life maps are colored to verify the effectiveness of the proposed algorithm. The experimental results show that the proposed EDDA algorithm can solve the four-color map problem with more than 100 regions.
... Another approach to minimizing the number of conflicts using metaheuristics is the use of population-based algorithms. They are often combined with tabu search or other local search mechanisms, which can be applied in a population-based algorithm in several ways, into so-called hybrid evolutionary algorithms, yielding some of the best recently developed coloring metaheuristics [30]. For instance, tabu search may be used as a mutation operator as in MACOL [24]; it may also simply be applied on each new population member to restore a decent level of conflict-freeness after crossover. ...
Preprint
We give an overview of the 2022 Computational Geometry Challenge targeting the problem Minimum Partition into Plane Subsets, which consists of partitioning a given set of line segments into a minimum number of non-crossing subsets.
Chapter
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Chapter
A typical combinatorial optimization problem, Graph Coloring Problem (GCP), has a wide range of applications in the fields of science and engineering. A cuckoo quantum evolutionary algorithm (CQEA) is proposed for the GCP, which is based on the framework of quantum-inspired evolutionary algorithm. To reduce iterations for the search of the chromatic number, the initial quantum population is generated with random initialization assisted by inheritance. Moreover, improvement of global exploration is achieved by incorporating the cuckoo search strategy, and a local search operation, as well as a perturbance strategy, is developed to enhance its performance on GCP. Numerical results show that CQEA has strong exploration and exploitation ability, and is competitive compared with the state-of-the-art heuristic algorithms.KeywordsGraph coloring problemQuantum-inspired evolutionary algorithmCuckoo search algorithm
Thesis
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