1. Discrete ABC Based on Similarity for GCP
Department of Computer Science, Graduate School of Systems and Information Engineering, University of Tsukuba, Japan
Kui Chen
Division of Information Engineering, Faculty of Engineering, Information and System, University of Tsukuba, Japan
Hitoshi Kanoh

2. Process of Presentation
Problem definition
Related work
Proposed method
Experiments
Conclusion
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3. Problem Definition
In this part, we define graph coloring problem and objective function
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4. Graph Coloring Problem
Graph coloring problem: given an undirected graph with vertices and edges, each vertex is assigned one of colors so that none of the vertices connected with an edge have the same color.
The coding scheme of GCP in our paper.
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5. Conflict of Graph Definition of conflict
For example, the conflict of the graph given in page 4 is 1.
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6. Definition of Objective Function
The objective function converges to optimal value 1 if there is no such two adjacent vertices that are assigned the same color.
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7. Related Work
In this part, we introduced original artificial bee colony algorithm and some related work on graph coloring problem.
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8. Original Artificial Bee Colony
Original artificial bee colony is an optimization algorithm in swarm intelligence. It consists of 3 bee groups:
Employed bees: explore the search space and discover food sources randomly.
Onlooker bees: select a food source with a probability according to its fitness and update the food source.
Scout bee: abandon a food source which cannot be improved over a predefined iteration number limit and generate a new food source randomly.
*Food sources: the candidate solutions of GCP.
*limit: the only parameter which should be set by user.
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9. Original Update Strategy
Original ABC’s update strategy
𝑖,𝑗∈{1, 2, 3, …, 𝑁} (N is swarm size)
𝑙∈{1, 2, 3,…, 𝑛} ( is vertices number)
𝑖≠𝑗
𝜙 is a uniform random number in [-1, 1]
The original update strategy is used to solve continuous optimization problems only, our goal is to discretize it directly without any hybrid algorithm.
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10. Related Work on GCP (1)
Takuya Aoki, PSO algorithm with transition probability based on hamming distance for graph coloring problem, 2015.
Problems: experiment on graph 3 coloring problems shows that HDPSO is an efficient method and obtains better result than GA. However, when the graph size becomes large, its performance turns to be worse.
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11. Related Work on GCP (2)
Iztok Fister Jr, A hybrid artificial bee colony algorithm for graph 3-coloring, 2012.
Problems: though hybrid ABC outperforms the original ABC, it is designed for solving graph 3-coloring problems only and difficult to be applied to other discrete optimization problems.
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12. Proposed Method
In this part, we give the proposed method: Discrete Artificial Bee Colony Based on Similarity.
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13. Definition of Similarity
The Similarity is defined as below:
: the Hamming distance between two solutions.
2. rn: a uniformly distributed random number between [0, 1].
3. n: the vertices number in a given graph.
4. Similarity: describes the similar degree between two solutions.
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14. Update Strategy – The Procedure
Procedure update-strategy(swarm) : for each in swarm : randomly select a solution calculate the Similarity between and generate a random number r in [0, 1] if r < Similarity : randomly choose u components from and replace the corresponding components of by them if fitness( ) > fitness( ) : replace by else : keep unchanged
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15. Update Strategy– Key Points
There are two key points in our update strategy:
Parameter u should not be too large. We found that if u is very large, the candidate solutions will become very similar and the diversity of swarm is lost. So we set the u value to a small integer.
Random number rn is used to extend the search range so that more candidate solutions can take part in the improvement of current solution.
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16. Main Procedure
Procedure proposed-ABC : initialization while (optimal solution is not found) and (loop time < max iteration times) : send employed bees (explore the search space) send onlooker bees (update candidate solutions and find optimal solution) send scout bee (abandon a solution which cannot be improved in limit loop times)
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17. Experiments
In this part, we design some experiments to compare our method with HDPSO and show the advantages of the proposed method.
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18. Constraint Density
The constraint density is the level of difficulty for a given graph. The rough relationship between constraint density and difficulty is below [Hogg, 1994]:
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19. Parameter Dependence – limit (1)
First of all, we must examine the dependence of the success rate on the parameters u and limit. To examine parameter limit, we use below settings:
Parameters
Value
Swarm size (N)
200
Graph size (n) 90
Constraint density (d)
2.5 (the most difficult problem)
Max iteration
10000 u 3
Run 50
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20. Parameter Dependence – limit (2)
The result is below:
The optimal limit value is 90.
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21. Parameter Dependence – u (1)
Then, we fix limit on its optimal value 90 and examine success rate on different u. The parameters settings are:
Parameters
Value
Swarm size (N)
200
Graph size (n) 90
Constraint density (d)
2.5 (the most difficult problem)
Max iteration
10000
limit
Run 50
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22. Parameter Dependence – u (2)
The result is below:
The optimal u value is 2.
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23. Comparative Study
The proposed method is compared with HDPSO on different constraint density to evaluate its performance. The two algorithms are compared according to two measures:
Success rate
Average evaluation times of objective function
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24. Parameters Settings Parameters Value Swarm size (N) 200 Graph size (n)
90, 120 and 150
Constraint density (d)
From 1.5 to 10
Max iteration
10000 u
2 (use the optimal value)
limit
90 (use the optimal value)
Run
100
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25. Success Rate (n = 90)
The result is below:
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26. Success Rate (n = 120)
The result is below:
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27. Success Rate (n = 150)
The result is below:
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28. Average Evaluation (n = 90)
The result is below:
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29. Average Evaluation (n = 120)
The result is below:
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30. Average Evaluation (n = 150)
The result is below:
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31. Conclusion
In the final part, we give our conclusion and the future work.
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32. Conclusion
We have introduce a new discrete ABC with Similarity. From the experiment, we find:
The success rate of the proposed ABC is much higher than HDPSO. For example, when n = 150, the success rate of proposed ABC and HDPSO are 20% and 1% respectively.
The proposed ABC is much faster than HDPSO. Even if the success rates are both 100% when d is larger than 4, the average evaluation of the proposed ABC are much lower than HDPSO.
The proposed ABC is effective and outperforms HDPSO dramatically.
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33. Future Work
The Similarity has been used in discrete PSO and ABC. So we think it may also be applied to discretize other swarm optimization algorithms.
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34. Acknowledgments
We wish to thank Dr. Claus Aranha of University of Tsukuba for his helpful comments and suggestions. This work is supported by JSPS KAKENHI Grant Number 15K00296.
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