An Approximation Algorithm for a Variant of Dominating Set Problem
Round 1
Reviewer 1 Report
==== SUMMARY ====
The paper is about a distributed algorithm for the total dominating set problem, that is, given an undirected graph G = (V, E), find a minimum size subset of nodes T that so that all the vertices (including those in T) have a neighbor in T.
The distributed version assume that each node contain an independant computer that can exchange information synchronously with the neighbors of the node.
The given algorithm is a constant-time approximation algorithm. More exaclty, the number of times some information is sent through the network is constant. However, the time complexity of each node is linear in the number of neighbors. (And the cummulative time complexity of all the nodes is linear in the number of edges of the graph).
The approximation ratio and the number of rounds depend on a parameter k.
==== MAIN REMARKS ====
The paper is clear and interesting. The english level seems adequate.
I had some difficulties to check all the technical proofs at the end of the paper. Some arguments seem missing or wrong. I admit I do not always understand how the authors get some of the results.
Particularly, from my point of view, some of the use of the dynamic degree in the proof should be the degree.
Please see the details below.
Also, due to the numerous notations, the algorithm is hard to follow.
I appreciated the given example but I think there could be a more efficient way to present it. I did some suggestions below.
Thus, I suggest a major revision mostly to update the proofs.
==== MAJOR REMARKS ====
Example of the algorithm :
If would be way more easier to read a table containing successive useful information than the current paragraph.
For instance, to explain how we go from figure 1a to figure 1b, I would prefer a table containing, for each node, the successive values of
xi, delta-tilde(vi), tau(2)(vi), (tau(2)(vi) + 1)^h/(h+1), a(vi) (at line 13), a(1)(vi), the new value of xi at line 17/18 and finally if the node becomes black or not at line 20/21/22
I would suggest to do this only of the first h, and only for the two first and the last values of m (so that it does not take too many space in the paper).
Take any other iteration if you judge that more appropriate.
Then put Figure 1 and remove the whole paragraph.
The proofs of Lemma 1, 2, 3 and Theorem 1 seem incomplete. I could not understand some of the arguments and I would prefer the authors to double check their proof and add some more arguments to explain these points.
I added some details below.
Proof of Lemma 1 :
I do not understand why setting xi to 1 if delta-tilde(xi) >= (Delta + 1)^(h/k) makes delta-tilde(xi) <= (Delta + 1)^(h/k) at the end of the iteration.
Page 7 : Proof of Lemma 2, L 153 : there are two verbs "is" in the sentence.
Proof of Lemma 3 :
I do not understand why zi increases only for the white vertices. A black vertice vi can have a neighbor vj for which the xj increases, in which case zi increases.
Line 173 : Why does the number of increased zi is at most (tau^(2)(vj) + 1)^(h/h+1). Since the increase of x is equally distributed among the neigbors, it should be the degree of vi
Please explain how you get line 179 ? (it this juste the result of inequalities (1) multiplied by a^(1)(vj)^(m/m+1) ?)
Proof of Th 1:
I do not see why every node is covered by the solution.
Why does the inequality at line 186 proves this ?
Should not the inequality after "in the outer loop iteration is at most" be the degree of vi instead of the dynamic degree ?
The inequality line 189 does not seem to be correct. It does not work for v8 on the figure 1a example.
I think there is an additionnal k in the definition of yj line 192.
===== MINOR REMARKS =====
Page 1 L 19 : the vertices represent (remonve the "s")
Page 1 L 21 22 : The sentence "Due to each server ... when a server crashes" should be placed after the definition of TDS and MTDS. Or at least an introductive sentence should explain what is a total dominating set.
Page 1 L 33 : What is a dynamic degree ? Is this notion defined page 3 ? An informal definition should come first.
Page 1 L 33 : Why starting the sentence with "On the other hand" ? There is no corresponding sentence with "On the one hand" (or any equivalent expression).
Page 1 L 37 : I would insist that, although the number of rounds is constant, the time complexity of each node is linear.
Page 2 L 54 : remove "where Delta is the maximum degree"
You should add some details to the following sentences.
Page 2 L 55 : "summarized some ... on some special graphs" : which results, which special graphs?
Page 2 L 57 58 : "summarized some ... some special graphs classes" : which classes?
Page 2 L 61 : "G[T] satisfies certain specific properties" : which ones ?
Page 2 L 65 : "or not in a class of graphs " which one ?
Page 2 L 69 : "For the minimal TDS problem" : is this a typo ? Do you mean the minimum TDS problem or the problem of finding a TDS from which no node can be removed ? In the first case, does that mean that the previous results were not associated to the Minimum TDS ?
Page 2 L 70 : what is an unfair central scheduler or an unfair distributed scheduler ?
Page 2 L 72 : in this context, does the running time means the number of rounds ?
Page 3 L 82 : the notion of dynamic degree is still not defined
Page 3 L 92 : tau^(2)(v_i) and \bar(h) are used in the formula but none of these notations were defined previously
Page 3 L 98 : there is an additional "use" at the beginning of the line
The algorithm does only give a relaxed solution.
How do you get the TDS from x ? You take only the xi that equal 1? I think it is never explained.
Lemma 1 and 2 : "we can obtain X is at most Y" ; this sentences seems uncorrect, I would just write "X <= Y"
Author Response
Please see the attachment
Author Response File: 
Author Response.pdf
Reviewer 2 Report
I found no errors in the article. The algorithm seems correct, and the claims are also correctly proven. I have only two following notes.
Line 78. δ(2)(vi) is the maximum degree of vi and vi in the 2-neighborhood. In the given case δ(2)(v6)=5= δ(v6) is not the maximum degree in the 2-neighborhood.
Line 101. Can you explain the meaning of k? In the given case, is k=5 specified or chosen? Can k=1?
Author Response
Please see the attachment
Author Response File: Author Response.pdf
Reviewer 3 Report
Overall paper presentation is good and the results seem correct. The similarity index is high than the standard index of similarity. please reduce it to below 19%. The overall presentation can be improved by checking spelling and grammar mistakes.
Author Response
Please see the attachment
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors clearly answered all the remarks. Particularly, I misunderstood the main result of the paper.
This paper is an approximation of a continuous relaxation of the min TDS problem. It is important to write this in the abstract and in the introduction. I appreciate that the authors wrote this in the conclusion but it should be placed at the beginning of the paper. Currently, the abstract and the introduction are misleading and a futur reader may think (as I did) that the algorithm is designed to solve the (integer) min TDS problem.
This lead me to my main remark :
what are the benefit of such an approximation while we can solve the continuous relaxation in polynomial time ? I would like that the authors emphasize these benefits in the introduction.
Minor remark
Just to be sure, here is how I understook the proof of Lemma 1 :
Either delta-tilde(vi) < Delta^h/k in which case the expected result is achieved or delta-tilde(vi) >= Delta^h/k. For this second case, as it is proved, between lines 175 and 176 of the paper, tau^(2)^(h / h+1) <= Delta^h/k, then tau^(2)^(h / h+1) <= delta-tilde(vi). Finally, according to lines 17-19, this means that, at the end of the loop, delta-tilde(vi) = 1, thus at the end of the loop delta-tilde(vi) <= Delta^h/k.
Is this correct ? if so, I suggest to rewrite the proof to add some few details. Particularly, the "Therefore we only need ..." at line 175 is counter intuitive.
Finally, for Lemma 3, to be sure that the definition of zi is clear, I suggest to rewrite the following sentence : "the increased xi are equally distributed to the zj of the vertices vj in N(vi), where the vertices vj are colored white before the increment of xi." using simply "the increased xi are equally distributed to the zj of the vertices in {vj \in N(vi) | vj is colored white before the increment of xi at line 17}"
If this is done correctly, I think the paper can be published. The new version with Table 1 make the algorithm clearer. And, now that I understand that the algorithm approximates the continuous relaxation, Theorem 1 seems way more correct.
Author Response
Please see the attachment, thanks.
Author Response File: Author Response.pdf
