A Novel Hybrid Algorithm for Minimum Total Dominating Set Problem
Round 1
Reviewer 1 Report
The authors propose a novel methodology to find the minimum total dominating set (MTDS) in a graph. The proposed approach exploits a mixture of evolutionary computation and local search. Overall the paper seems good. The topic is nicely introduced, the methodology is properly explained, and the results seem convincing (with a few major issues, see below).
Major points
When you provide background on genetic algorithms (GA), there are basically no citations. You should provide references at least for genetic algorithms. Also, the term "memetic algorithms" has been defined in a paper by Moscato. See:
- Moscato, P., 1989. On evolution, search, optimization, genetic algorithms and martial arts: Towards memetic algorithms. Caltech concurrent computation program, C3P Report, 826, p.1989.
- Moscato, P., Cotta, C. and Mendes, A., 2004. Memetic algorithms. In New optimization techniques in engineering (pp. 53-85). Springer, Berlin, Heidelberg.
You should probably cite that, as it is a fundamental citation in the domain.
During the experimental results, how do you you find the "better solutions"? The issue is that you are comparing the results of stochastic processes, so even if the averages are different, there is still the standard deviation to take into account, and the two sets of results might be indistinguishable. You should run at least 30 repetitions for each stochastic method, and then compare the results using a Kolmogorov-Smirnov two-sample test. KS tries to test the hypotheses that the two sets of points are taken from two different distributions, within a certain confidence. Only using that, you can actually estimate that one set of results is "better".
Also, the reader might not be familiar with the DIMACS benchmark. For each instance, in Table 1, could you add the number of vertexes and edges? As you previously talked about large graphs, it would be interesting to know exactly how large each instance is. There might even be a correlation between size and performance of the greedy algorithm.
Ideas for future works: finding the MTDS is surely interesting, but if you relax the problem, it could be conceived as a multi-objective problem. For example, what if you could find a MTDS that is *almost* correct, but is missing the adjacency of one vertex? In this case, the problem could have two objectives: minimize number of points in a candidate set, maximize number of adjacencies to vertexes. It could then be interesting to explore the Pareto front resulting from this approach. Evolutionary algorithms are particularly good in multi-objective optimization, see the NSGA2 for example.
Minor points
page 1, line 29: The evolutionary algorithm -> I would say "Evolutionary algorithms include". Also, it's "evolutionary programming"
Author Response
1) When you provide background on genetic algorithms (GA), there are basically no citations. You should provide references at least for genetic algorithms. Also, the term "memetic algorithms" has been defined in a paper by Moscato. See:
- Moscato, P., 1989. On evolution, search, optimization, genetic algorithms and martial arts: Towards memetic algorithms. Caltech concurrent computation program, C3P Report, 826, p.1989.
- Moscato, P., Cotta, C. and Mendes, A., 2004. Memetic algorithms. In New optimization techniques in engineering (pp. 53-85). Springer, Berlin, Heidelberg.
You should probably cite that, as it is a fundamental citation in the domain..
Response：
Thank you very much for your suggestions. We have modified our paper accordingly and cite the above two papers in references 17 and 18.2) During the experimental results, how do you you find the "better solutions"? The issue is that you are comparing the results of stochastic processes, so even if the averages are different, there is still the standard deviation to take into account, and the two sets of results might be indistinguishable. You should run at least 30 repetitions for each stochastic method, and then compare the results using a Kolmogorov-Smirnov two-sample test. KS tries to test the hypotheses that the two sets of points are taken from two different distributions, within a certain confidence. Only using that, you can actually estimate that one set of results is "better".
Response：
Thank you very much for your suggestions. We have modified the paper accordingly. For our problem, We use box-plot to give the performance of the algorithm more intuitively. The DIMACS benchmark is divided into 10 groups. We choose 10 instance from the 10 groups and every instance is run 10 times independently. The box-plot shows the distribution of the total dominating set values of massive graphs. The flatter the box-plot, the smaller the variance. From Figure 1 we can see that our algorithm performs better than ACO.
3) Also, the reader might not be familiar with the DIMACS benchmark. For each instance, in Table 1, could you add the number of vertexes and edges? As you previously talked about large graphs, it would be interesting to know exactly how large each instance is. There might even be a correlation between size and performance of the greedy algorithm.
Response:
Thank you for your suggestions. We have added two columns in Table 1 to introduce the number of edges and vertices of DIMACS benchmark.
4) Ideas for future works: finding the MTDS is surely interesting, but if you relax the problem, it could be conceived as a multi-objective problem. For example, what if you could find a MTDS that is *almost* correct, but is missing the adjacency of one vertex? In this case, the problem could have two objectives: minimize number of points in a candidate set, maximize number of adjacencies to vertexes. It could then be interesting to explore the Pareto front resulting from this approach. Evolutionary algorithms are particularly good in multi-objective optimization, see the NSGA2 for example.
Response:
Thank you for your suggestions. We have considered these problems and modified the paper accordingly.
5) Minor points
page 1, line 29: The evolutionary algorithm -> I would say "Evolutionary algorithms include". Also, it's "evolutionary programming"
Response:
Thank you for your suggestions. We have modified the paper accordingly.
Reviewer 2 Report
In this paper, the authors develop a new hybrid evolutionary algorithm tailored for
the numerical treatment of the minimum total dominating set (MTDS) problem.
The algorithm is tested against to other methods on several instances.
On the one hand, all seems to be technically correct (though the related work
part should be discussed in more detail) and the algorithm does what is is supposed
to do. On the other hand, I have several hesitations regarding this work
- the choice of the journal puzzles me a bit. Apart from the problem description I
do not see any maths, in particular in the proposed material.
- I am not an expert in MTDS problems, however, a quick internet search reveals
that there are many algorithms addressing this problem. In the paper, only one
specialized algorithm has been chosen (while ACO is a general purpose algorithm). See
for instance the links below, but there are some other interesting links that can be
found quickly.
https://www.researchgate.net/publication/225636181_Experimental_Analysis_of_Heuristic_Algorithms_for_the_Dominating_Set_Problem
https://www.maxwell.vrac.puc-rio.br/34169/34169.PDF
- the overall innovation is ok for a conference paper, not sure if this is enough for
a journal publication. For this, I expect a more in-depth discussion of the proposed
algorithm.
Author Response
1) - the choice of the journal puzzles me a bit. Apart from the problem description I
do not see any maths, in particular in the proposed material.
Response：
Thank you very much for your suggestions. Our paper is submitted for the special issue “Evolutionary Computation” of Mathematics. So this paper mainly described an evolutionary algorithm to solve the MTDS problem, which is a NP-hard algorithm.
2) I am not an expert in MTDS problems, however, a quick internet search reveals
that there are many algorithms addressing this problem. In the paper, only one
specialized algorithm has been chosen (while ACO is a general purpose algorithm). See
for instance the links below, but there are some other interesting links that can be
found quickly.
https://www.researchgate.net/publication/225636181_Experimental_Analysis_of_Heuristic_Algorithms_for_the_Dominating_Set_Problem
https://www.maxwell.vrac.puc-rio.br/34169/34169.PDF
Response:
Thank you for your suggestions. There are many algorithms to solve the minimum dominant set problem, but there are almost no algorithms to solve MTDS problem. We are the first to use heuristic algorithm to sole MTDS problem. ACO is an effective algorithm for solving minimum dominant set problem [1][2]. And as the reviewer said, ACO is a general purpose algorithm. Therefore, it is easy to use ACO to solve MTDS, and the experimental results show that ACO works well. So we choose ACO as our main comparison algorithm.
[1] Jovanovic R., Tuba M., Simian D. Ant colony optimization applied to minimum weight dominating set problem[C]. Proc. of the 12th WSEAS International Conference on Automatic Control, Modelling & Simulation (ACMOS). World Scientific and Engineering Academy and Society, Stevens Point, USA, 2010, 322-326.
[2] Potluri A, Singh A. Hybrid metaheuristic algorithms for minimum weight dominating set[J]. Applied Soft Computing, 2013, 13(1): 76-88.
3) the overall innovation is ok for a conference paper, not sure if this is enough for
a journal publication. For this, I expect a more in-depth discussion of the proposed
algorithm.
Response:
Thank you for your suggestions. For further analyzing the efficiency of the algorithm, we have added some box-plots. The DIMACS benchmark is divided into 10 groups. We choose 1 instance from every group. Every instance is run 10 times independently. We draw box-plots based on the results. The box-plot shows the distribution of the total dominating set values of massive graphs. The flatter the box-plot, the smaller the variance. From Figure 1 we can see that our algorithm performs better than ACO.
Round 2
Reviewer 1 Report
The authors replied to my remarks. Still, there are several issues left:
You decided to run 10 instances 10 times each (recommended is all instances, 30 repetitions).
You did not run a statistical test on the results, so we still do not know whether the result you found is actually better or just the same as ACO.
I am not sure that Figure 1 actually always proves that your method is better. Overall, it seems to have lower standard deviation, but in most cases the results seem indistinguishable from ACO.
Without knowing whether the results you obtained are actually better or worse than ACO (using a statistical test on the two distributions of results you obtained), the paper cannot be published in its current form. I would recommend you run the experiments and the statistical tests.
Author Response
You decided to run 10 instances 10 times each (recommended is all instances, 30 repetitions).
Response：
Thanks for your kindly suggestion. We have already rerun all experiments and modified the relevant parts.
You did not run a statistical test on the results, so we still do not know whether the result you found is actually better or just the same as ACO.
I am not sure that Figure 1 actually always proves that your method is better. Overall, it seems to have lower standard deviation, but in most cases the results seem indistinguishable from ACO.
Without knowing whether the results you obtained are actually better or worse than ACO (using a statistical test on the two distributions of results you obtained), the paper cannot be published in its current form. I would recommend you run the experiments and the statistical tests.
Response：
According to the reviewer’s suggestion, we have used Kolmogorov-Smirnov test to verify the performance our algorithm and our competitor.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The authors have addressed all my comments. I think the paper
can now be published in its current form.
Author Response
The authors have addressed all my comments. I think the paper
can now be published in its current form.
Response：
Thank you.
Round 3
Reviewer 1 Report
The authors answered all my questions appropriately, and they proved that their method performs better than the ACO they used for comparison. The manuscript can be published in its present form.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
Marked up manuscript
Comments for author File: Comments.pdf
Author Response
Thank you very much for your suggestions, we have modified the paper accordingly.
Author Response File: Author Response.pdf
Reviewer 2 Report
Authors presented the idea of solving the minimum total dominating set using hybrid solution (combining GA with LS).
Authors must improve the paper before it can be published.
a)is not the proposed algorithm the result of Baldwin effect? Local search as an additional algorithm in GA is interpreted as a Baldwin effect, the authors should know that and write it in the paper.
b) the mathematical model was not presented in the work, but only the pseudocode was described. Please add an exact (full) mathematical model,
c) where is the statistical analysis? where are the time charts?
d) it would be good to add comparisons with other algorithms
e) Why was GA chosen instead of another algorithm? Test the operation of other heuristic algorithms instead of GA and describe it at work
f) Papers older than 4 years can be boldly removed from the bibliography and replaced with new ones. It is worth paying attention to the following ones
-The evolutionary cost of baldwin effect in the routing and spectrum allocation problem in elastic optical networks
-Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space, Applied Sciences
g)What is the computational complexity of this algorithm?
Author Response
1) is not the proposed algorithm the result of Baldwin effect? Local search as an additional algorithm in GA is interpreted as a Baldwin effect, the authors should know that and write it in the paper.
Response：
Thank you very much for your suggestions. We added a comparison algorithm GRASP, which
is a classical local search algorithm for combinatorial optimization problems. The experimental
results show that our algorithm performs much better than GRASP and greedy algorithms. So,
GA also plays an important role in HELG.
2) the mathematical model was not presented in the work, but only the pseudocode was described. Please add an exact (full) mathematical model,
Response:
The mathematical model of our algorithm is based on a classical hybrid evolutionary algorithm MEMETIC, which takes advantage of local search framework as well as population-based search. We described it in our paper.
3) where is the statistical analysis? where are the time charts?
Response:
Thank you for your suggestions. We have analyzed the results of the experiment in more detail and modified the table.
4) it would be good to add comparisons with other algorithms
Response:
Thank you very much for your suggestions. We added a greedy randomized adaptive search procedure (GRASP), which is a classical local search algorithm for combinatorial optimization problems as our new comparison algorithm.
5) Why was GA chosen instead of another algorithm? Test the operation of other heuristic algorithms instead of GA and describe it at work
Response:
GA is very efficient and widely used to solve some optimization problems. Thus, we consider using GA to solve combinatorial optimization problems, especially to solve MTDS for better results. We use a classical local search algorithm GRASP to solve MTDS as a comparison algorithm, which doesn’t use GA. The experimental results show that our algorithm combining GA and LS obtains better results.
6) Papers older than 4 years can be boldly removed from the bibliography and replaced with new ones. It is worth paying attention to the following one
-The evolutionary cost of baldwin effect in the routing and spectrum allocation problem in elastic optical networks
-Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space, Applied Sciences
Response:
Thank you very much. We modified the references and read the two articles which are very meaningful, and thus we added them to the reference. They are located in references 14 and 15, respectively.
7) What is the computational complexity of this algorithm?
Response:
We have already given the computational complexity of our proposed algorithm.
Author Response File: Author Response.pdf
Reviewer 3 Report
Primarily the article is experimental. However, what algorithms are being compared is stated in an unclear manner. The experimental design requires significant improvement. This article is suitable for soft computing. Writing needs improvement. Some of the suggestions are given below. Very little is described about exact/approximation-algorithm. More detailed introduction is required. The justification for soft-computing must be more sound.
P1.13 vertex in V/DS
must be:
vertex in V\DS
P1.21 function, which is a good property in mathematics.
A desirable property.
P1.20 with approximation ratio ln(\delta - 0.5) + 1.5 [5].
What is \delta here?
P1.22 However, when the size of problem increases or the problem is difficult to calculate [6–9], the
23 approximation algorithm will become invalid.
The phrase "problem is difficult to calculate" is grammatically incorrect.
The phrase "approximation algorithm will become invalid" is an incorrect statement.
P1.23 Under these circumstances, heuristic algorithms [10–13]
24 are used by researchers.Although
There needs to be a space between sentences.
P2.56 The remaining sections are arranged as follows: in Sect.2, we give some necessary information. In
57 Sect.3, we introduce the novel scoring heuristic. The evolution algorithm HELG for MTDS is described
58 in Sect.4.
There must be space between Sec. and the number following it.
P2. 52. We carry out some experiments to compare HELG with a greedy algorithm for MTDS on DIMACS [16]
benchmark.
- What greedy algorithm is used in unclear.
- [16] Does not pose Dominating Set problem. They might just be using dataset specified for another problem.
P3.93 algorithm. The algorithm including three important phases: generation of a population including n
What is "n"? How is it related to the problem?
Author Response
1) P1.13 vertex in V/DS must be: vertex in V\DS
2) P1.21 function, which is a good property in mathematics. A desirable property.
3) P1.20 with approximation ratio ln(\delta - 0.5) + 1.5 [5]. What is \delta here?
4) P1.22 However, when the size of problem increases or the problem is difficult to calculate [6–9], the approximation algorithm will become invalid.
The phrase "problem is difficult to calculate" is grammatically incorrect.
The phrase "approximation algorithm will become invalid" is an incorrect statement.
5) P1.23 Under these circumstances, heuristic algorithms [10–13] 24 are used by researchers. Although There needs to be a space between sentences.
6) P2.56 The remaining sections are arranged as follows: in Sect.2, we give some necessary information. In 57 Sect.3, we introduce the novel scoring heuristic. The evolution algorithm HELG for MTDS is described 58 in Sect.4. There must be space between Sec. and the number following it.
Response:
Thank you very much for your suggestions. We have modified the paper accordingly.
For suggestion 3), the \delta refers to the maximum degree of the given graph.
7) P2. 52. We carry out some experiments to compare HELG with a greedy algorithm for MTDS on DIMACS [16] benchmark.
- What greedy algorithm is used in unclear.
- [16] Does not pose Dominating Set problem. They might just be using dataset specified for another problem
Response:
We have modified the paper accordingly and described the greedy algorithm in more detail.
The DIMACS benchmark is a classical benchmark for graph algorithms. It comes from the Second DIMACS Implementation Challenge. It is often used in the graph algorithms to compare and evaluate the efficiency of the algorithms.
8) P3.93 algorithm. The algorithm including three important phases: generation of a population including n
What is "n"? How is it related to the problem?
Response:
“n” is a parameter used in our algorithm. It is the number of initial populations. The value of it is given in Section 5.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Many of my suggestions have been incorporated into the new version. The manuscript has not been improved much, but is acceptable from my point of view. Let's see, what the other referees and the editor think.
Author Response
Thank you very much. We have modified the paper accordingly.
Reviewer 2 Report
There is still no decent mathematical model. There is the pseudocode but this is a scientific work, not a project documentation.
The results should be supported by comparative tests and statistical analysis - please add graphs of time depending on iteration. How do variables in the algorithm affect effectiveness?
Author Response
The results should be supported by comparative tests and statistical analysis - please add graphs of time depending on iteration. How do variables in the algorithm affect effectiveness?
Response：
Thank you very much for your suggestions, we have modified the paper accordingly.
we show the time-to-target plot in Figure 1 to compare HELG with GRASP on san200_0.9_3.clq.mis.cnf and its target 27. To obtain the plot, we performed 100 independent runs of each algorithm on san200_0.9_3.clq.mis.cnf. The results show that HELG performs better than GRASP.
Author Response File: 
Author Response.docx
Reviewer 3 Report
English must still be improved. Some instances are shown below. Some of these instances seems to arise from a lack of understanding. Genetic algorithms are used from early 1990s. The relevance to a math. journal must be justified by showing a theoretical result. Experimental results that cannot be quantified do not suffice. Some instances where writing can be improved are shown below. This article belongs to soft-computing/AI/Natural-computing. It is a misfit here.
P1.
25 algorithms cannot guarantee the optimality of the solution they find ,
Remove extra space.
P1.
23 However, when the size of problem increases [6–9], the approximation algorithm will be invalid.
The above statement is unclear.
P2.
40 Recently, evolutionary algorithms play an important role in solving optimization problems [14,15].
Must be rewritten.
P2.
The remaining sections are arranged as follows: in Sect. 2 , we give some necessary information.
P5.
The corresponding scoring values are updated in line 12. Subsequently, the algorithm selects a
135 vertex from V/CS with the highest score and adds it into CS(lines 13-14). After that, the cost value of
136 each v 2 V/CS
Incorrect symbols are used. Need space after CS.
P6
190 instance, the GRASP also runs 10 times independently with different random seeds, until the time
191 limit (100s) is satisfied.
In the remaining 9 instances, HELG gets the same minimum
199 solutions with the greedy algorithm. Compared with the GRASP, HELG can obtain better minimum
200 solutions in 48 instances of 61 instances. In the remaining 13 instances, HELG gets the same minimum
201 size with GRASP. Among them, HELG gets better average values than GRASP in 7 instances, and gets
202 the same average value with GRASP but performs faster in 6 instances.
Conclusion:
HELG performs well in solving MIDS.
MIDS must be MTDS.
Author Response
P1.
25 algorithms cannot guarantee the optimality of the solution they find ,
Remove extra space.
P1.
23 However, when the size of problem increases [6–9], the approximation algorithm will be invalid.
The above statement is unclear.
P2.
40 Recently, evolutionary algorithms play an important role in solving optimization problems [14,15].
Must be rewritten.
P2.
The remaining sections are arranged as follows: in Sect. 2 , we give some necessary information.
P5.
The corresponding scoring values are updated in line 12. Subsequently, the algorithm selects a
135 vertex from V/CS with the highest score and adds it into CS(lines 13-14). After that, the cost value of
136 each v 2 V/CS
Incorrect symbols are used. Need space after CS.
P6
190 instance, the GRASP also runs 10 times independently with different random seeds, until the time
191 limit (100s) is satisfied.
In the remaining 9 instances, HELG gets the same minimum
199 solutions with the greedy algorithm. Compared with the GRASP, HELG can obtain better minimum
200 solutions in 48 instances of 61 instances. In the remaining 13 instances, HELG gets the same minimum
201 size with GRASP. Among them, HELG gets better average values than GRASP in 7 instances, and gets
202 the same average value with GRASP but performs faster in 6 instances.
Conclusion:
HELG performs well in solving MIDS.
MIDS must be MTDS.
Response：
Thank you very much for your suggestions. We have modified the paper accordingly.
Author Response File: Author Response.docx
Round 3
Reviewer 2 Report
In my opinion, the mathematical model is still not improved and not readable.
The analysis of the results is very poor.
The summary still lacks a description of why the results are good.
Reviewer 3 Report
Some writing errors:
P1
However, in real life and industrial production, the size of problem is very large. when the size
24 of problem increases [6–9], the approximation algorithm will be invalid.
P2
Dawid Połap et al. propose three proposition
P2
The remaining sections are arranged as follows: in Sect. 2, we give some necessary information.
What information?
P4
The corresponding scoring values are updated in line 12. Subsequently, the algorithm
142 selects a vertex from VnDS with the highest score and adds it into CS (lines 13-14).
Adds it to?
P8
in at most 26.68 and 53.7
216 seconds, considerably longer than HELG.
Add respectively.
