A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
The field of designing subexponential time parameterized algorithms has gained a lot of momentum lately. While the subexponential time algorithm for Planar-k-Path (finding a path of length at least k on planar graphs) has been known for last 15 years. There was no such algorithms known on directed planar graphs. In this talk, I will survey this journey of designing subexponential time parameterized algorithms for finding a path of length at least k in planar undirected graphs to planar directed graphs; highlighting the new tools and techniques that got developed on the way.
Swarm Intelligence Heuristics for Graph Coloring ProblemMario Pavone
In this research work we present two novel swarm
heuristics based respectively on the ants and bees artificial
colonies, called AS-GCP and ABC-GCP. The first is based
mainly on the combination of Greedy Partitioning Crossover
(GPX), and a local search approach that interact with the
pheromone trails system; the last, instead, has as strengths
three evolutionary operators, such as a mutation operator; an
improved version of GPX and a Temperature mechanism. The
aim of this work is to evaluate the efficiency and robustness of both developed swarm heuristics, in order to solve the classical Graph Coloring Problem (GCP). Many experiments have been performed in order to study what is the real contribution of variants and novelty designed both in AS-GCP and ABC-GCP.
A first study has been conducted with the purpose for setting
the best parameters tuning, and analyze the running time
for both algorithms. Once done that, both swarm heuristics
have been compared with 15 different algorithms using the
classical DIMACS benchmark. Inspecting all the experiments
done is possible to say that AS-GCP and ABC-GCP are very
competitive with all compared algorithms, demonstrating thus
the goodness of the variants and novelty designed. Moreover,
focusing only on the comparison among AS-GCP and ABCGCP is possible to claim that, albeit both seem to be suitable to solve the GCP, they show different features: AS-GCP presents a quickly convergence towards good solutions, reaching often the best coloring; ABC-GCP, instead, shows performances more robust, mainly in graphs with a more dense, and complex topology. Finally, ABC-GCP in the overall has showed to be more competitive with all compared algorithms than AS-GCP as average of the best colors found.
A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
The field of designing subexponential time parameterized algorithms has gained a lot of momentum lately. While the subexponential time algorithm for Planar-k-Path (finding a path of length at least k on planar graphs) has been known for last 15 years. There was no such algorithms known on directed planar graphs. In this talk, I will survey this journey of designing subexponential time parameterized algorithms for finding a path of length at least k in planar undirected graphs to planar directed graphs; highlighting the new tools and techniques that got developed on the way.
Swarm Intelligence Heuristics for Graph Coloring ProblemMario Pavone
In this research work we present two novel swarm
heuristics based respectively on the ants and bees artificial
colonies, called AS-GCP and ABC-GCP. The first is based
mainly on the combination of Greedy Partitioning Crossover
(GPX), and a local search approach that interact with the
pheromone trails system; the last, instead, has as strengths
three evolutionary operators, such as a mutation operator; an
improved version of GPX and a Temperature mechanism. The
aim of this work is to evaluate the efficiency and robustness of both developed swarm heuristics, in order to solve the classical Graph Coloring Problem (GCP). Many experiments have been performed in order to study what is the real contribution of variants and novelty designed both in AS-GCP and ABC-GCP.
A first study has been conducted with the purpose for setting
the best parameters tuning, and analyze the running time
for both algorithms. Once done that, both swarm heuristics
have been compared with 15 different algorithms using the
classical DIMACS benchmark. Inspecting all the experiments
done is possible to say that AS-GCP and ABC-GCP are very
competitive with all compared algorithms, demonstrating thus
the goodness of the variants and novelty designed. Moreover,
focusing only on the comparison among AS-GCP and ABCGCP is possible to claim that, albeit both seem to be suitable to solve the GCP, they show different features: AS-GCP presents a quickly convergence towards good solutions, reaching often the best coloring; ABC-GCP, instead, shows performances more robust, mainly in graphs with a more dense, and complex topology. Finally, ABC-GCP in the overall has showed to be more competitive with all compared algorithms than AS-GCP as average of the best colors found.
Multiscale Gradient Based – Directional CFA Interpolation with RefinementIJTET Journal
Abstract—Single sensor digital cameras capture only one color value for every pixel location. The process of reconstructing a full color image from these incomplete color samples output from an image sensor overlaid with a color filter array (CFA) is called demosaicing or Color Filter Array (CFA) interpolation. The most commonly used CFA configuration is the Bayer filter. The proposed demosaicing method makes use of multiscale color gradients to adaptively combine color difference estimates from horizontal and vertical directions and determine the contribution of each direction to the green channel interpolation. This method does not require any thresholds and is non iterative. The red and blue channels are then refined using structural approximation.
An overview of Rademacher Averages, a fundamental concept from statistical learning theory that can be used to derive uniform sample-dependent bounds to the deviation of samples averages from their expectations.
Designing a pencil beam pattern with low sidelobesPiyush Kashyap
In this paper, a system has been designed for an operational frequency of 1.27 GHz consisting of an 8 element array of parasitic dipoles illuminated by a 4 element center fed array of active dipoles with Dolph-Chebyshev excitation coefficients. The array is designed to achieve a fairly pencil beam pattern suitable for direction of arrival estimation purposes. Array geometry and configuration is optimized for both active and parasitic elements using the PSO tool in FEKO. A directive radiation pattern is obtained with a gain of 14.5 dBi in the broadside direction along with a beamwidth of 30.29o. VSWR of 1.58 is achieved. Further, an iterative least square valued error estimation approach using phase control to achieve a desired array factor pattern for an n-element linear array, has been shown to be effective for larger number of iterations. The array excitation coefficients achieved were consistent with the Dolph-Chebyshev coefficients used in our antenna array design. With the ability to introduce nulls and steering the main beam in desired directions along with a pencil beam radiation pattern, beamsteering has been illustrated and the MUSIC algorithm for direction of arrival estimation has been implemented
A Firefly Algorithm for Optimizing Spur Gear Parameters Under Non-Lubricated ...irjes
Firefly algorithm is one of the emerging evolutionary approaches for complex and non-linear
optimization problems. It is inspired by natural firefly‟s behavior such as movement of fireflies based on
brightness and by overcoming the constraints such as light absorption, obstacles, distance, etc. In this research,
firefly‟s movement had been simulated computationally to identify the best parameters for spur gear pair by
considering the design and manufacturing constraints. The proposed algorithm was tested with the traditional
design parameters and found the results are at par in less computational time by satisfying the constraints.
Multiscale Gradient Based – Directional CFA Interpolation with RefinementIJTET Journal
Abstract—Single sensor digital cameras capture only one color value for every pixel location. The process of reconstructing a full color image from these incomplete color samples output from an image sensor overlaid with a color filter array (CFA) is called demosaicing or Color Filter Array (CFA) interpolation. The most commonly used CFA configuration is the Bayer filter. The proposed demosaicing method makes use of multiscale color gradients to adaptively combine color difference estimates from horizontal and vertical directions and determine the contribution of each direction to the green channel interpolation. This method does not require any thresholds and is non iterative. The red and blue channels are then refined using structural approximation.
An overview of Rademacher Averages, a fundamental concept from statistical learning theory that can be used to derive uniform sample-dependent bounds to the deviation of samples averages from their expectations.
Designing a pencil beam pattern with low sidelobesPiyush Kashyap
In this paper, a system has been designed for an operational frequency of 1.27 GHz consisting of an 8 element array of parasitic dipoles illuminated by a 4 element center fed array of active dipoles with Dolph-Chebyshev excitation coefficients. The array is designed to achieve a fairly pencil beam pattern suitable for direction of arrival estimation purposes. Array geometry and configuration is optimized for both active and parasitic elements using the PSO tool in FEKO. A directive radiation pattern is obtained with a gain of 14.5 dBi in the broadside direction along with a beamwidth of 30.29o. VSWR of 1.58 is achieved. Further, an iterative least square valued error estimation approach using phase control to achieve a desired array factor pattern for an n-element linear array, has been shown to be effective for larger number of iterations. The array excitation coefficients achieved were consistent with the Dolph-Chebyshev coefficients used in our antenna array design. With the ability to introduce nulls and steering the main beam in desired directions along with a pencil beam radiation pattern, beamsteering has been illustrated and the MUSIC algorithm for direction of arrival estimation has been implemented
A Firefly Algorithm for Optimizing Spur Gear Parameters Under Non-Lubricated ...irjes
Firefly algorithm is one of the emerging evolutionary approaches for complex and non-linear
optimization problems. It is inspired by natural firefly‟s behavior such as movement of fireflies based on
brightness and by overcoming the constraints such as light absorption, obstacles, distance, etc. In this research,
firefly‟s movement had been simulated computationally to identify the best parameters for spur gear pair by
considering the design and manufacturing constraints. The proposed algorithm was tested with the traditional
design parameters and found the results are at par in less computational time by satisfying the constraints.
Improved Firefly Algorithm for Unconstrained Optimization ProblemsEditor IJCATR
in this paper, an improved firefly algorithm with chaos (IFCH) is presented for solving unconstrained optimization
problems. Several numerical simulation results show that the algorithm offers an efficient way to solve unconstrained optimization
problems, and has a high convergence rate, high accuracy and robustness.
Parallel hybrid chicken swarm optimization for solving the quadratic assignme...IJECEIAES
In this research, we intend to suggest a new method based on a parallel hybrid chicken swarm optimization (PHCSO) by integrating the constructive procedure of GRASP and an effective modified version of Tabu search. In this vein, the goal of this adaptation is straightforward about the fact of preventing the stagnation of the research. Furthermore, the proposed contribution looks at providing an optimal trade-off between the two key components of bio-inspired metaheuristics: local intensification and global diversification, which affect the efficiency of our proposed algorithm and the choice of the dependent parameters. Moreover, the pragmatic results of exhaustive experiments were promising while applying our algorithm on diverse QAPLIB instances . Finally, we briefly highlight perspectives for further research.
Genetic Algorithm for the Traveling Salesman Problem using Sequential Constru...CSCJournals
This paper develops a new crossover operator, Sequential Constructive crossover (SCX), for a genetic algorithm that generates high quality solutions to the Traveling Salesman Problem (TSP). The sequential constructive crossover operator constructs an offspring from a pair of parents using better edges on the basis of their values that may be present in the parents' structure maintaining the sequence of nodes in the parent chromosomes. The efficiency of the SCX is compared as against some existing crossover operators; namely, edge recombination crossover (ERX) and generalized N-point crossover (GNX) for some benchmark TSPLIB instances. Experimental results show that the new crossover operator is better than the ERX and GNX.
This presentation is about applications of graph theory applications....it is updated version it was given at international conference at applications of graph theory at KAULALAMPUR MALYSIA 2OO7
The Effect of Updating the Local Pheromone on ACS Performance using Fuzzy Log...IJECEIAES
Fuzzy Logic Controller (FLC) has become one of the most frequently utilised algorithms to adapt the metaheuristics parameters as an artificial intelligence technique. In this paper, the 휉 parameter of Ant Colony System (ACS) algorithm is adapted by the use of FLC, and its behaviour is studied during this adaptation. The proposed approach is compared with the standard ACS algorithm. Computational results are done based on a library of sample instances for the Traveling Salesman Problem (TSPLIB).
An analysis between different algorithms for the graph vertex coloring problem IJECEIAES
This research focuses on an analysis of different algorithms for the graph vertex coloring problem. Some approaches to solving the problem are discussed. Moreover, some studies for the problem and several methods for its solution are analyzed as well. An exact algorithm (using the backtracking method) is presented. The complexity analysis of the algorithm is discussed. Determining the average execution time of the exact algorithm is consistent with the multitasking mode of the operating system. This algorithm generates optimal solutions for all studied graphs. In addition, two heuristic algorithms for solving the graph vertex coloring problem are used as well. The results show that the exact algorithm can be used to solve the graph vertex coloring problem for small graphs with 30-35 vertices. For half of the graphs, all three algorithms have found the optimal solutions. The suboptimal solutions generated by the approximate algorithms are identical in terms of the number of colors needed to color the corresponding graphs. The results show that the linear increase in the number of vertices and edges of the analyzed graphs causes a linear increase in the number of colors needed to color these graphs.
Behavior study of entropy in a digital image through an iterative algorithmijscmcj
Image segmentation is a critical step in computer vision tasks constituting an essential issue for pattern recognition and visual interpretation. In this paper, we study the behavior of entropy in digital images through an iterative algorithm of mean shift filtering. The order of a digital image in gray levels is defined. The behavior of Shannon entropy is analyzed and then compared, taking into account the number of iterations of our algorithm, with the maximum entropy that could be achieved under the same order. The use of equivalence classes it induced, which allow us to interpret entropy as a hyper-surface in real m dimensional space. The difference of the maximum entropy of order n and the entropy of the image is used to group the the iterations, in order to caractrizes the performance of the algorithm.
An Intelligent Method for Accelerating the Convergence of Different Versions ...aciijournal
In a wide range of applications, solving the linear system of equations Ax = b is appeared. One of the best
methods to solve the large sparse asymmetric linear systems is the simplified generalized minimal residual
(SGMRES(m)) method. Also, some improved versions of SGMRES(m) exist: SGMRES-E(m, k) and
SGMRES-DR(m, k). In this paper, an intelligent heuristic method for accelerating the convergence of three
methods SGMRES(m), SGMRES-E(m, k), and SGMRES-DR(m, k) is proposed. The numerical results
obtained from implementation of the proposed approach on several University of Florida standard
matrixes confirm the efficiency of the proposed method.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A COMPARISON BETWEEN SWARM INTELLIGENCE ALGORITHMS FOR ROUTING PROBLEMSecij
Travelling salesman problem (TSP) is a most popular combinatorial routing problem, belongs to the class of NP-hard problems. Many approacheshave been proposed for TSP.Among them, swarm intelligence (SI) algorithms can effectively achieve optimal tours with the minimum lengths and attempt to avoid trapping in local minima points. The transcendence of each SI is depended on the nature of the problem. In our studies, there has been yet no any article, which had compared the performance of SI algorithms for TSP perfectly. In this paper,four common SI algorithms are used to solve TSP, in order to compare the performance of SI algorithms for the TSP problem. These algorithms include genetic algorithm, particle swarm optimization, ant colony optimization, and artificial bee colony. For each SI, the various parameters and operators were tested, and the best values were selected for it. Experiments oversome benchmarks fromTSPLIBshow that
artificial bee colony algorithm is the best one among the fourSI-basedmethods to solverouting problems like TSP.
Color Image Segmentation Based On Principal Component Analysis With Applicati...CSCJournals
In this paper we propose a segmentation method for multi-spectral images in the HSV space, based on the Principal Component Analysis to generate grayscale images. Then the Firefly Algorithm has been applied on the gray-level images in a histogram-based research of cluster centroids. The FA is a metaheuristic optimization algorithm, centered on the flashing behaviour of fireflies. The Firefly Algorithm is performed to determine automatically the number of clusters and to select the gray levels for partitioning pixels into homogeneous regions. Successively, these gray values are employed during the initialization step of a Gaussian Mixture Model for estimation of parameters, evaluated through the Expectation-Maximization technique. The coefficients of the linear super-position of Gaussians can be regarded as the prior probabilities of each component. Applying the Bayes rule, the posterior probabilities have been estimated and their maxima are used to assign each pixel to the clusters, according to their gray values.
International Refereed Journal of Engineering and Science (IRJES) is a peer reviewed online journal for professionals and researchers in the field of computer science. The main aim is to resolve emerging and outstanding problems revealed by recent social and technological change. IJRES provides the platform for the researchers to present and evaluate their work from both theoretical and technical aspects and to share their views.
www.irjes.com
Research on Chaotic Firefly Algorithm and the Application in Optimal Reactive...TELKOMNIKA JOURNAL
Firefly algorithm (FA) is a newly proposed swarm intelligence optimization algorithm. Like many other general swarm intelligence optimization algorithms, the original version of FA is easy to trap into local optima. In order to overcome this drawback, the chaotic firefly algorithm (CFA) is proposed. The methods of chaos initialization, chaos population regeneration and linear decreasing inertia weight have been introduced into the original version of FA so as to increase its global search mobility for robust global optimization. The CFA is calculated in Matlab and is examined on six benchmark functions. In order to evaluate the engineering application of the algorithm, the reactive power optimization problem in IEEE 30 bus system is solved by CFA. The outcomes show that the CFA has better performance compared to the original version of FA and PSO.
Similar to Memetic Firefly algorithm for combinatorial optimization (20)
Cuckoo Search Algorithm: An IntroductionXin-She Yang
This presentation explains the fundamental ideas of the standard Cuckoo Search (CS) algorithm, which also contains the links to the free Matlab codes at Mathswork file exchanges and the animations of numerical simulations (video at Youtube). An example of multi-objective cuckoo search (MOCS) is also given with link to the Matlab code.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
Online aptitude test management system project report.pdfKamal Acharya
The purpose of on-line aptitude test system is to take online test in an efficient manner and no time wasting for checking the paper. The main objective of on-line aptitude test system is to efficiently evaluate the candidate thoroughly through a fully automated system that not only saves lot of time but also gives fast results. For students they give papers according to their convenience and time and there is no need of using extra thing like paper, pen etc. This can be used in educational institutions as well as in corporate world. Can be used anywhere any time as it is a web based application (user Location doesn’t matter). No restriction that examiner has to be present when the candidate takes the test.
Every time when lecturers/professors need to conduct examinations they have to sit down think about the questions and then create a whole new set of questions for each and every exam. In some cases the professor may want to give an open book online exam that is the student can take the exam any time anywhere, but the student might have to answer the questions in a limited time period. The professor may want to change the sequence of questions for every student. The problem that a student has is whenever a date for the exam is declared the student has to take it and there is no way he can take it at some other time. This project will create an interface for the examiner to create and store questions in a repository. It will also create an interface for the student to take examinations at his convenience and the questions and/or exams may be timed. Thereby creating an application which can be used by examiners and examinee’s simultaneously.
Examination System is very useful for Teachers/Professors. As in the teaching profession, you are responsible for writing question papers. In the conventional method, you write the question paper on paper, keep question papers separate from answers and all this information you have to keep in a locker to avoid unauthorized access. Using the Examination System you can create a question paper and everything will be written to a single exam file in encrypted format. You can set the General and Administrator password to avoid unauthorized access to your question paper. Every time you start the examination, the program shuffles all the questions and selects them randomly from the database, which reduces the chances of memorizing the questions.
2. re
y algorithm for
combinatorial optimization
Iztok Fister Jr., Xin-She Yang,y Iztok Fister,z and Janez Brestx
Abstract
Fire
y algorithms belong to modern meta-heuristic algorithms inspired by nature that can be
successfully applied to continuous optimization problems. In this paper, we have been applied the
3. re
y algorithm, hybridized with local search heuristic, to combinatorial optimization problems,
where we use graph 3-coloring problems as test benchmarks. The results of the proposed memetic
4. re
y algorithm (MFFA) were compared with the results of the Hybrid Evolutionary Algorithm
(HEA), Tabucol, and the evolutionary algorithm with SAWmethod (EA-SAW) by coloring the suite
of medium-scaled random graphs (graphs with 500 vertices) generated using the Culberson random
graph generator. The results of
5. re
y algorithm were very promising and showed a potential that
this algorithm could successfully be applied in near future to the other combinatorial optimization
problems as well.
To cite paper as follows: I. Fister Jr., X.-S. Yang, I. Fister, J. Brest, Memetic
6. re
y algo-
rithm for combinatorial optimization, in Bioinspired Optimization Methods and their Applications
(BIOMA 2012), B. Filipic and J.Silc, Eds. Jozef Stefan Institute, Ljubljana, Slovenia, 2012
1 arXiv:1204.5165v2 [math.OC] 10 May 2012
University of Maribor, Faculty of electrical engineering and computer science Smetanova 17, 2000 Maribor;
Electronic address: iztok.
7. ster@guest.arnes.si
yUniversity of Cambridge, Department of Engineering Trumpington Street, Cambridge CB2 1PZ, UK; Elec-tronic
address: xy227@cam.ac.uk
zUniversity of Maribor, Faculty of electrical engineering and computer science Smetanova 17, 2000 Maribor;
Electronic address: iztok.
8. ster@uni-mb.si
xUniversity of Maribor, Faculty of electrical engineering and computer science Smetanova 17, 2000 Maribor;
Electronic address: janez.brest@uni-mb.si
9. I. INTRODUCTION
Nature, especially biological systems, has always been an inspiration for those scientists
who would like to transform some successful features of a biological system into computer
algorithms for ecient problem solving. Birds, insects, ants, and
10. sh may display some
so-called, collective or swarm intelligence in foraging, defending, path
11. nding, etc. This
collective intelligence of self-organizing systems or agents has served as a basis for many
good and ecient algorithms developed in the past, e.g.: ant-colony optimization [7], particle
swarm optimization[22], arti
12. cial bee colony [11, 12, 21], bacterial foraging [23]. Today, all
these algorithms are referred to as swarm intelligence.
The
13. re
y algorithm (FFA) belongs to swarm intelligence as well. It was developed by
X. S. Yang [28]. This algorithm is based on the behavior of
18. re
ies are, the more attractive they will seem to other. In FFA, each
19. re
y represents a point in a search space. When the attractiveness is proportional to the
objective function the search space is explored by moving the
20. re
ies towards more attractive
neighbors.
FFA has displayed promising results when applied to continuous optimization prob-lems
[29, 30]. Conversely, within the area of combinatorial optimization problems, only
a few papers have been published to date. Therefore, aim of this paper is to show that
FFA can be applied to this kind of optimization problems as well. In this context, FFA for
graph 3-coloring (3-GCP) has been developed. 3-GCP can informally be de
21. ned as follows:
How to color a graph G with three colors so that none of the vertices connected with an
edge is colored with the same color. The problem is NP-complete as proved by Garey and
Johnson [16].
The most natural way to solve this problem is in a greedy fashion. Here, the vertices are
ordered into a permutation and colored sequentially one after the other. However, the quality
of coloring depends on the order in which the vertices will be colored. For example, the naive
method orders the vertices of graph randomly. One of the best traditional heuristics for graph
coloring today is DSatur by Brelaz [2], which orders the vertices v according to saturation
degree (v). The saturation degree denotes the number of distinctly colored vertices to the
2
22. vertex v.
This problem cannot be solved by an exact algorithm for graph instances of more than
100 vertices. Therefore, many heuristic methods have been developed for larger instances
of graphs. These methods can be divided into local search [15] and hybrid algorithms [26].
One of the more successful local search heuristic was Tabucol developed by Herz and De
Werra [19], who employed the tabu search proposed by Glover [17]. The most eective local
search algorithms today are based on reactive partial local search [1, 25], adaptive mem-ory
[20], and variable search space [1]. On the other hand, various evolutionary algorithms
have been hybridized using these local search heuristics. Let us refer to three such algorithms
only: the hybrid genetic algorithm by Fleurent and Ferland [13], the hybrid evolutionary
algorithm by Galinier and Hao [14], and the memetic algorithm for graph coloring by Lu
and Hao [24].
Some modi
23. cations of the original algorithm need to be performed in order to apply FFA
to 3-GCP. The original FFA algorithm operates on real-valued vectors. On the other hand,
the most traditional heuristics act on the permutation of vertices. In order to incorporate the
bene
24. ts of both, solutions of the proposed memetic FFA (MFFA) algorithm are represented
as real-valued vectors. The elements of these vectors represent weigths that determine how
hard the vertices are to color. The higher the weight is, the sooner the vertex should be
colored. The permutation of vertices is obtained by sorting the vertices according to their
weights. The DSatur traditional heuristic is used for construction of 3-coloring from this
permutation. A similar approach was used in the evolutionary algorithm with the SAW
method (EA-SAW) of Eiben et al. [8], and by the hybrid self-adaptive dierential evolution
and hybrid arti
25. cial bee colony algorithm of Fister et al. [10, 11]. Additionally, the heuristical
swap local search is incorporated into the proposed MFFA. In order to preserve the current
best solution in the population, the elitism is considered by this algorithm.
The results of the proposed MFFA algorithm for 3-GCP were compared with the results
obtained with EA-SAW, Tabucol, and HEA by solving an extensive set of random medium-scale
graphs generated by the Culberson graph generator [6]. The comparison between these
algorithms shows that the results of the proposed MFFA algorithm are comparable, if not
better, than the results of other algorithms used in the experiments.
The structure of this paper is as follows: In Section 2, the 3-GCP is discussed, in detail.
The MFFA is described in Section 3, while the experiments and results are presented in
3
26. Section 4. The paper is concluded with a discussion about the quality of the results, and
the directions for further work are outlined.
II. GRAPH 3-COLORING
Let us suppose, an undirect graph G = (V;E) is given, where V is a set of vertices v 2 V
for i = 1; : : : ; n, and E denotes a set of edges that associate each edge e 2 E for i = 1; : : : ;m
to the unordered pair e = fvi; vjg for i = 1; : : : ; n and j = 1; : : : ; n. Then, the vertex
3-coloring (3-GCP) is de
27. ned as a mapping c : V ! S, where S = f1; 2; 3g is a set of three
colors and c a function that assigns one of the three colors to each vertex of G. A coloring
s is proper if each of the two vertices connected with an edge are colored with a dierent
color.
3-GCP can be formally de
28. ned as a constraint satisfaction problem (CSP). It is rep-resented
as a pair hS; i, where S denotes the search space, in which all solutions s 2 S
are feasible, and a Boolean function on S (also a feasibility condition) that divides the
search space into feasible and unfeasible regions. To each e 2 E the constraint be is as-signed
with be(hs1; : : : ; sni) = true if and only if e = fvi; vjg and si6= sj . Suppose that
Bi = fbeje = fvi; vjg ^ j = 1 : : :mg de
29. nes the set of constraints belonging to variable
vi. Then, the feasibility condition is expressed as a conjunction of all the constraints
(s) = ^v2V Bv(s).
As in evolutionary computation, constraints can be handled indirectly in the sense of a
penalty function, that punishes the unfeasible solutions. The farther the unfeasible solution
is from the feasible region, the higher is the penalty function. The penalty function is
expressed as:
f(s) = min
Xn
i=0
(s;Bi); (1)
where the function (s;Bi) is de
30. ned as:
(s;Bi) =
8
1 if s violates at least one be 2 Bi;
0 otherwise:
:
(2)
Note that Eq. (1) also represents the objective function. On the other hand, the same
4
31. equation can be used as a feasibility condition in the sense that (s) = true if and only if
f(s) = 0. If this condition is satis
32. ed a proper graph 3-coloring is found.
III. MEMETIC FIREFLY ALGORITHM FOR GRAPH 3-COLORING
The phenomenon of
34. re
ies
ash their lights that can be admired on clear
summer nights. This light is produced by a complicated set of chemical reactions. Fire
y
ashes in order to attract mating partners and serve as a protection mechanism for warning
o potential predators. Their light intensity I decreases when the distance r from the light
source increases according to term I _ r2. On the other hand, air absorbs the light as the
distance from the source increases.
When the
ashing light is proportional to the objective function of the problem being
optimized (i.e., I(w) _ f(w)), where w represents the candidate solution) this behavior
of
38. re
ies are unisex, their attractiveness is proportional to their
brightness, and the brightness of a
39. re
y is aected or determined by the landscape of the
objective function.
These rules represent the basis on which the
40. re
y algorithm acts [28]. The FFA algorithm
is population-based, where each solution denotes a point in the search space. The proposed
MFFA algorithm is hybridized with a local search heuristic. In this algorithm, the solution is
represented as a real-valued vector wi = (wi;1; : : : ;wi;n) for i = 1 : : :NP, where NP denotes
the size of population P. The vector wi determines the weights assigned to the corresponding
vertices. The values of the weights are taken from the interval wi;j 2 [lb; ub], where lb and ub
are the lower and upper bounds, respectively. The weights represent an initial permutation
of vertices (vi). This permutation serves as an input to the DSatur heuristic that obtains
the graph 3-coloring. The pseudo-code of the MFFA algorithm is illustrated in Algorithm 1.
As can be seen from Algorithm 1, the search process of the MFFA algorithm (statements
within the while loop) that is controlled by generation counter t consists of the following
functions:
EvaluateFFA(): evaluating the solution. This evaluation is divided into two parts:
In the
41. rst part, the solution wi is transformed into a permutation of vertices (vi)
according to the non-decreasing values of the weights. In the second part, the permu-
5
42. Algorithm 1 Pseudo code of the MFFA algorithm
1: t = 0; fe = 0; found = FALSE; s = ;;
2: P(t) = InitializeFFA();
3: while (!TerminateFFA(fe, found)) do
4: fe += EvaluateFFA(P(t));
5: P0 = OrderFFA(P(t));
6: found = FindTheBestFFA(P(t); s);
7: Pt+1 = MoveFFA(Pt; P0);
8: t = t+1;
9: end while
tation of vertices (vi) is decoded into a 3-coloring si by the DSatur heuristic. Note
that the 3-coloring si represents the solution of 3-GCP in its original problem space,
where the quality of the solution is evaluated according to Eq. (1). Conversely, looking
for a new solution is performed within the real-valued search space.
OrderFFA(): forming an intermediate population P0 by copying the solutions from
the original population P(t) and sorting P(t) according to the non-decreasing values of
the objective function.
FindTheBestFFA(): determining the best solution in the population P(t). If the best
solution in P(t) is worse than the s the later replaces the best solution in P(t), otherwise
the former becomes the best solution found so far s.
MoveFFA(): moving the
43. re
ies towards the search space according to the attractive-ness
of their neighbour's solutions.
Two features need to be developed before this search process can take place: the initializa-tion
(function InitializeFFA()) and termination (function TerminateFFA()). The population
is initialized randomly according to the following equation:
wi;j = (ub lb) rand(0; 1) + lb; (3)
where the function rand(0; 1) denotes the random value from the interval [0; 1]. The process
is terminated when the
45. ed: the number of objec-tive
function evaluations fe reaches the maximum number of objective function evaluations
(MAX FES) or the proper coloring is found (found == true).
6
55. 0e
r2
i;j ),
while the latter is the randomized move in the search space (determined by the randomized
parameter ). Furthermore, the attractiveness depends on the
56. 0 that is the attractiveness at
r = 0, absorbtion coecient
, and Euclidian distance between the attracted and attracting
57. re
y ri;j .
A. The heuristical swap local search
After the evaluation step of the MFFA algorithm, the heuristical swap local search tries
to improve the current solution. This heuristic is executed until the improvements are
detected. The operation of this operator is illustrated in Fig. 1, which deals with a solution
on G with 10 vertices. This solution is composed of a permutation of vertices v, 3-coloring
s, weights x, and saturation degrees . The heuristical swap unary operator takes the
58. rst
uncolored vertex in a solution and orders the predecessors according to the saturation degree
descending. The uncolored vertex is swapped with the vertex that has the highest saturation
degree. In the case of a tie, the operator randomly selects a vertex among the vertices with
higher saturation degrees (1-opt neighborhood).
0 1 2 3 4 5 6 7 8
1 2 2 1 3 0 1 2 0
.2 .4 .7 .9 .5 .3 .1 .8 .6
3 2 2 3 1 0 3 2 0
0 1 2 5 4 3 6 7 8
.2 .4 .7 .3 .5 .9 .1 .8 .6
swap
FIG. 1: The heuristical swap unary operator
In Fig. 1, an element of the solution corresponding to the
59. rst uncolored vertex 5 is in
dark gray and the vertices 0 and 3 with the highest saturation degree are in light gray. From
7
60. vertices 0 and 3, heuristical swap randomly selects vertex 3 and swaps it with vertex 5 (the
right-hand side of Fig. 1).
IV. RESULTS
In the experimental work, the results of the proposed MFFA algorithm were compared
with the results of: EA-SAW, HEA, and Tabucol. The algorithms used in the experiments
were not selected coincidentally. In order to help the developers of new graph coloring
algorithms, the authors Chiarandini and Stutzle [4] made the code of Tabucol and HEA
available within an online compendium [5]. On the other hand, this study is based on
the paper of Eiben and al. [8], in which the authors proposed the evolutionary algorithm
with SAW method for 3-GCP. The source code of this algorithm can also be obtained from
Internet [18]. The goal of this experimental work was see whether the MFFA could also be
applied to combinatorial optimization problems like 3-GCP.
The characteristics of the MFFA in the experiments were as follows. The population
size was set at 500. The MAX FES was
61. xed at 300,000 by all algorithms to make this
comparison as fair as possible. All algorithms executed each graph instance 25 times. The
algorithm parameters of MFFA were set as follows: = 0:1,
62. 0 = 0:1, and
= 0:8. Note
that these values of algorithm parameters optimize the performance of MFFA and were
obtained during parameter tuning within the extensive experimental work. This parameter
tuning satis
64. rst perspective of parameter tuning, as proposed by Eiben and Smit [9],
i.e., choosing parameter values that optimize the algorithm's performance. In order to satisfy
the second perspective of the parameter tuning, i.e., how the MFFA performance depends
on its parameter values, only the in
uence of the edge density was examined because of
limited paper length.
The algorithms were compared according to the measures success rate (SR) and average
evaluations of objective function to solution (AES). While the
65. rst measure represents
the ratio between the number of successfully runs and all runs, the second determines the
eciency of a particular algorithm. The aim of these preliminary experiments was to show
that MFFA could be applied for 3-GCP. Therefore, any comparison of algorithms according
to the time complexity was omitted.
8
66. A. Test-suite
The test-suite, considered in the experiments, consisted of graphs generated with the
Culberson random graph generator [6]. The graphs generated by this generator are distin-guished
according to type, number of vertices n, edge probability p, and seeds of random
number generator q. Three types of graphs were employed in the experiments: uniform
(random graphs without variability in sizes of color sets), equi-partite, and
at. The edge
probabilities were varied in the interval p 2 0:008 : : : 0:028 with a step of 0.001. Finally, the
seeds were varied in interval q 2 1 : : : 10 with a step of one. As a result, 32110 = 630 dif-ferent
graphs were obtained. That is, each algorithm was executed 15; 750 times to complete
this experimental setup.
The experimental setup was selected so that a phase transition was captured. The phase
transition is a phenomenon that accompanies almost all NP-hard problems and determines
the region where the NP-hard problem passes over the state of solvability to a state
of unsolvability and vice versa [31]. Typically, this region is characterized by ascertain
problem parameter. This parameter is edge probability for 3-GCP. Many authors have
determined this region dierently. For example, Petford and Welsh [27] stated that this
phenomenon occurs when 2pn=3 16=3, Cheeseman et al. [3] when 2m=n 5:4, and
Eiben et al. [8] when 7=n p 8=n. In the presented case, the phase transition needed
to be by p = 0:016 over Petford and Welsh, by p 0:016 over Cheeseman, and between
0:014 p 0:016 over Eiben et al.
B. In
uence of the edge density
During this experiment, the in
uence of the edge density on the performance of the tested
algorithms were investigated. The results are illustrated in Fig. 2. This
67. gure consists of six
diagrams corresponding to graphs of dierent types (uniform, equi-partite, and
at), and
according to the measures SR and AES. In these diagrams, the average of those values
accumulated after 25 runs are presented. Especially, we focus on the behavior of tested
algorithms within the phase transition.
As can be seen in Fig. 2.a and Fig. 2.c, the results of MFFA outperformed the results
of the other algorithms according to the measure SR on uniform and equi-partite graphs.
9
68. 1.0
0.8
0.6
0.4
0.2
0.0
0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028
Edge probability
EA-SAW
Tabucol
HEA
MFFA
(a)SR by uniform graphs
3e+005
2e+005
2e+005
2e+005
1e+005
5e+004
0e+000
0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028
Edge probability
EA-SAW
Tabucol
HEA
MFFA
(b)AES by uniform graphs
1.0
0.8
0.6
0.4
0.2
0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028
Edge probability
EA-SAW
Tabucol
HEA
MFFA
(c)SR by equi-partite graphs
3e+005
2e+005
2e+005
2e+005
1e+005
5e+004
0e+000
0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028
Edge probability
EA-SAW
Tabucol
HEA
MFFA
(d)AES by equi-partite graphs
1.0
0.8
0.6
0.4
0.2
0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028
Edge probability
EA-SAW
Tabucol
HEA
MFFA
(e)SR by
at graphs
3e+005
2e+005
2e+005
2e+005
1e+005
5e+004
0e+000
0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028
Edge probability
EA-SAW
Tabucol
HEA
MFFA
(f)AES by
at graphs
FIG. 2: Results of algorithms for 3-GCP solving dierent types of random graphs
HEA was slightly better than Tabocol during the phase transition (p 2 [0:014; 0:016]), whilst
EA-SAW exposed the worst results within this region. As can be seen in Fig. 2.e,
at graphs
were the hardest nuts to crack for all algorithms. Here, the results of MFFA according to
measure SR were slightly worse but comparable to the results of HEA and Tabucol, whilst
EA-SAW reported the worst results.
According to the measure AES (Fig. 2.b and Fig. 2.d), the best results were reported for
MFFA by coloring the uniform and equi-partite graphs. On average, MFFA found solutions
using a minimum number of evaluations. Note that the highest peak by p = 0:014 denotes
the hardest instance of uniform and equi-partite graphs to color for almost all the observed
algorithms. Conversely, the graph with p = 0:015 was the hardest instance when coloring
10
69. the
at graphs.
In summary, the proposed MFFA outperformed the results of HEA and Tabucol when
coloring the uniform and equi-partite graphs, while by coloring the
at graphs it behaved
slightly worse. The results of EA-SAW fell behind the results of the other tested algorithms
for coloring all other types of graphs.
C. Discussion
The results of other parameter tuning experiments have been omitted because of limited
paper length. Notwithstanding, almost four characteristics of MFFA could be exposed from
the results of the last experiment. First, it is very important whether the movement of
i-th
70. re
y according to Eq. (4) is calculated from the position of the j-th
71. re
y taken from
the population P(t) or the intermediate population P0, because the former incorporates
an additional randomness within the search process. Thus, the results were signi
72. cantly
improved. Second, an exploration of the search space depends on the best solution in the
population that directs the search process to more promising regions of the search space.
As a result, this elitist solution needs to be preserved. Third, the local search serves as
a search mechanism for detailed exploration of the basins of attraction. Consequently,
those solutions that would normally be ignored by the regular FFA search process can be
discovered. Fourth, the parameter determines the size of the randomness move within
the search space. Unfortunately, all conducted tests to self-adapt this parameter did not
bear any improvement, at this moment. In summary, these preliminary results of MFFA
encourage us to continue developing this algorithm in the future.
V. CONCLUSION
The results of MFFA showed that this algorithm could in future be a very promising
tool for solving 3-GCP and, consequently, the other combinatorial optimization problems as
well. In fact, it produced better results than HEA and Tabucol, when coloring the medium-scale
uniform and equi-partite graphs. Unfortunately, this algorithm is slightly worse on
at
graphs that remains the hardest to color for all the tested algorithms.
However, these good results could be misleading until further experiments on large-scale
11
73. graphs (graphs with 1,000 vertices) are performed. Fortunately, in the sense of preserving
the obtained results, we have several ways for improving this MFFA in the future.
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14