Latency-Bounded Target Set Selection in Signed Networks
Round 1
Reviewer 1 Report
Traditional perfect target set selection (PTSS) is as follows:
(1) Pick some initially active vertices in a graph.
(2) While there exists an inactive vertex $v$ whose number of active neighbors exceeds a certain threshold, active $v$. Do this until no more vertices can be activated.
(3) The question is: What is the minimum number of initially active vertices needed to activate all vertices at the end?
This paper considers a variant of traditional PTSS by allowing a neighbor of a vertex $v$ to have a negative impact on $v$.
E.g., a vertex may tend to be inactive when one of its neighbors is activated.
The considered variant is reasonable because people may have enemies.
Then the paper gives inapproximability results via reductions and also approximation algorithms for some special cases.
Because the model is natural, I would suggest acceptance/minor revision.
Minor comments:
P.~1: ``Epidemiology''->``epidemiology'' (global comment)
P.~1: The first sentence of Sec.~1 is hard to read
P.~1: ``their state''->``their states''
P.~2: ``each currently unaware neighbor v of u changes its state to informed with probability equals to''->grammer error?
Line~56, p.~2: I think the footnote should appear after the period? (global comment)
P.~2: ``it has been shown a polynomial-time algorithm ...''->weird
P.~2: ``... as to latency-bounded diffusion processes''->weird
Lines~116--117, p.~3: Too many occurrences of ``with''
Lines~118--119, p.~3: ``positive active''->``positively active'' and similarly for negative activation
Lines~143--144, p.~4: ``people decisions''->``people's decisions''
Line~192, p.~5: ``smaller than $\theta(u)$''->``less than $\theta(u)$''
Eq.~(1), p.~6: Use $\cdots$ instead of $\ldots$
Line~203, p.~6: ``All ... weights equals ...''->``All ... weights equal ...''
Line~368, p.~10: Use $\cdots$ to skip operations done over many objects and $\ldots$ for plain enumeration (so if we take $\land$'s, use $\cdots$)
Line~427, p.~12: ``an $(1+\log n)$-approximate solution''->``a $(1+\log n)$-approximate solution''
Line~492, p.~14: Using the hyphen as em dash looks weird (global comment)
Author Response
First of all, we want to thank you for your careful (and fast!) work on our paper, and for your comments that do helped us to improve it.
According to all reviewers' comments, all our changes are minor ones (typos, os some sintax mistake, or a required figure, or something like that) but the following two slightly more than minor ones:
1) pag. 7, line 243: we are showing that we can consider weakly positive boolean formulas in which at least one clause is positive *without loss of generality*. In the submitted version, the proof of this fact was longer than needed, since it is sufficient to notice that instances in which all clauses are negative do trivially admit a solution of size 0 (and, hence, they are solvable in polynomial-time)
2) pag. 12, three lines beneath line 416: the proof that S(E_i) \subseteq S(E_{i+1} is a direct consequence of E_i \subseteq E_{i+1}, as outlined in this new version. In the submitted version the proof was long and involved.
In what follows, please find your remarks and our answers. For the sake of completeness, in order you have access to all our changes, after that we also include the remarks from the other reviewers as well as our answers.
YOUR REMARKS:
1) ``Epidemiology''->``epidemiology'' (global comment)
ANSWER: done
2) The first sentence of Sec.~1 is hard to read: modified first sentence in Sec. 1
ANSWER: done. The original sentence "Diffusion processes are a widely investigated complex networks related topic aiming at analyzing how local interactions among nearby nodes may lead to the diffusion, possibly over the whole network, of some feature (opinion/information/disease) starting from a limited set of \emph{initiator} (or \emph{seed}) nodes. " has been modified to "The study of diffusion processes is a widely investigated topic in the complex networks setting. It aims
at analyzing how local interactions among nearby nodes may lead to the diffusion, possibly over the whole network, of some feature (opinion/information/disease) starting from a limited set of \emph{initiator} (or \emph{seed}) nodes. "
3) ``their state''->``their states''
ANSWER: done
4) ``each currently unaware neighbor v of u changes its state to informed with probability equals to''->grammer error?
ANSWER: replaced "equals" by "equal" - we hope this was the problem
5) Line~56, p.~2: I think the footnote should appear after the period? (global comment)
ANSWER: the two footnotes have been removed and their content has been moved within the text
6) P.~2: ``it has been shown a polynomial-time algorithm ...''->weird
ANSWER: the sentence has been modified to "... a polynomial-time algorithm spreading a feature over a set of required size within a constant number of time steps in graphs of bounded clique-width has been proposed"
7) P.~2: ``... as to latency-bounded diffusion processes''->weird
ANSWER: sentence changed to "in this paper, diffusion processes with an available fixed number of time steps within which informing all the nodes will be called {\em latency-bounded} diffusion processes"
8) Lines~116--117, p.~3: Too many occurrences of ``with''
ANSWER: sentence changed to "if $v$ is activated by $u$ then the probability that $v$ gets informed with the same orientation of $u$ is $\varphi_{u,v}$, and the probability that $v$ gets informed with the opposite orientation is $1-\varphi_{u,v}$"
9) Lines~118--119, p.~3: ``positive active''->``positively active'' and similarly for negative activation
ANSWER: done
10) Lines~143--144, p.~4: ``people decisions''->``people's decisions''
ANSWER: done
11) Line~192, p.~5: ``smaller than $\theta(u)$''->``less than $\theta(u)$''
ANSWER: done
12) Eq.~(1), p.~6: Use $\cdots$ instead of $\ldots$
ANSWER: done (in Eq. 1 and in the next formula)
13) Line~203, p.~6: ``All ... weights equals ...''->``All ... weights equal ...''
ANSWER: done
14) Line~368, p.~10: Use $\cdots$ to skip operations done over many objects and $\ldots$ for plain enumeration (so if we take $\land$'s, use $\cdots$)
ANSWER: done, tried to fix anywhere (thanks!)
15)Line~427, p.~12: ``an $(1+\log n)$-approximate solution''->``a $(1+\log n)$-approximate solution''
ANSWER: done
16) Line~492, p.~14: Using the hyphen as em dash looks weird (global comment)
ANSWER: fixed the two points in Sec. Conclusions and open problems (not found elsewhere)
REMARKS FROM REVIEWER 2:
0) "it reminds me of another problem called the Graph Burning problem (this might even be worth mentioning somewhere in the paper if it is not already) with more facets attached to it, where the model studied in your paper can be used to model many more things such as social networks and diffusion. If applicable, perhaps citing one of these papers may be relevant for the paper; though, I do not think it is necessary as the literature review of this paper is generally comprehensive. I think the results are interesting; however, my opinion might not be representative of an expert opinion for these types of optimization problems."
ANSWER: Yes, there are several analogies between diffusion processes and epidemics spreading (or what you call Graph burning). In this last setting our "negative influence" translates in agents contrasting the disease / fire spreading. Actually, there is a huge amount of literature concerning the so-called Firefighter problem. Our model (and our results) can be read also within the firefighter setting, as well as most of the literature related to diffusion processes is also suitable to describe epidemiological processes. We have decided to describe our model in the diffusion processes setting because it looks more natural. And adding a comparison to the literature about firefighting would have resulted really too long. So, we have decided to follow the same approach of the diffusion literature we have cited: disregarding the other side.
1) Page 1, Abstract, line 1: Maybe be a bit more specific here, some information does not spread via social networks; perhaps just rephrasing this sentence may make it read better.
ANSWER: done
2) Page 1, Abstract, line 4: perhaps, add the word "problem" after "Target Set Selection". This is done later in the abstract.
ANSWER: actually, the sentence is "A widely studied problem in the area of viral marketing is the Target Set Selection" and adding "problem" at the end would sound repetitive. Maybe we could say "Target Set Selection one", but it does not sound so fine
3) Page 1, Abstract, line 7: What is "it" in this sentence? Maybe being a bit more specific here will be helpful for the reader.
ANSWER: done
4) Page 1, Abstract, line 10: What is "first" in this context? I've read the paper and I do not think writing this helps the reader understand the restriction; maybe rephrasing this sentence may help. It does not need to be specific on what the restriction is, but "first" might not mean much here.
ANSWER: removed
5) Page 1, Section 1, line 14: Maybe capitalize the "p" in "processes"; perhaps also add a hyphen between "complex" and "networks".
ANSWER: the word "Diffusion" is capitalized just because it is the first word in the sentence If it were in the middle of a sentence, we would have written "diffusion processes". It is not the name of a topic, it is just a noun with its adjective. Writing "complex-networks" is more unusual than "complex networks": see, for instance, the entry in Wikipedia, or the name of the related conference (Complex networks, indeed), or even, more remarkably, Easley-Kleinberg's book https://www.cs.cornell.edu/home/kleinber/networks-book/networks-book.pdf and the literature cited therein.
6) Page 2, Section 1, line 33: Should the word "directed" be added between "edge-weighted" and "graph"? Somewhere here this should be clarified as later references may change in how they are interpreted by if the graph is a directed graph or not.
ANSWER: generally speaking, the graphs are directed, and we have added "directed" according to your suggestion. Since an undirected graph is a particular directed graph (in which arcs (u,v) and (v,u) both exists or both does not exist), this does not change the way the (rather informal) discussion of the results in the literature presented in the introduction can be interpreted.
7) Page 2, Section 1, line 66: 16 steps? If so, write "16 steps" in the parentheses. I was unsure here.
ANSWER: done
8) Page 3, Section 1, line 116: Add a comma between u and v so that it reads "if $v$ is activated by $u$, $v$ gets...".
ANSWER: comma added (although the sentence has been modified according to a different reviewer)
9) Page 4, Section 2, line 159 and line 163: Several times you use the phrases like Min-Horn Deletion-complete and Set Cover-hard. This is not as common of phrasing, so I suggest you write out once exactly what you mean by this; I'd recommend when you first write Min Horn Deletion-complete.
ANSWER: added a footnote and clarified the sentence
10) Page 4, Section 2, line 162: Maybe change 1-target set -> 1-Target Set; sometimes k-Target Set is used, other times k-target set is used. I suggest picking one and sticking with it. For example, this is done again on line 169 and also on Page 5 on line 184.
ANSWER: done
11) Page 5, Section 3: Maybe add a figure that shows a signed graph, and is used to illustrate the definitions for each +-version; in particular, the definition of $d^+(V',u)$ and $\pi^+(u)$.
ANSWER: done
12) Page 5, Section 3, line 191 (7 lines down): When you write "length" and "shortest", I suggest being more careful here. For example, in the number of arcs? Either way, being more precise here for both "maximum length" and "shortest positive path" will help the reader.
ANSWER: in graph theory, a directed/undirected path is defined as a sequence of arcs/edges (see, just to be easy, Wikipedia, but there is plenty of documents in the network) and, hence, the length of a path is defined as the number of arcs/edges composing it. The Shortest Path problem is defined accordingly (and on this definition Dijkstra's algorithm is based). Since it is the standard definition, we just refer to it (being more specific would sound redundant).
13) Page 5, Section 3, line 192 (one line down): Several times in the paper you use the set of natural numbers, I suggest either changing these to $\mathbb{Z}^+$ or being more explicit on the set of natural numbers (and which interpretation of them you are using). You use the natural numbers denoted this way many times, and some people might assume you are including 0 in, while others might not.
ANSWER: done
14) Page 6, Section 3, line 200: I recommend explicitly defining the size of a target set (the one we want as small as possible).
ANSWER: well, the size of a set is, trivially, its cardinality. We have added (a line above) the sentence "Needless to say, the size of a target set $I_0 \subseteq V$ is $|I_0|$."
15) Page 6, Section 4, line 220: Here you might want to add a sentence stating that at times you will say x_i \in c_j to mean x_i is a variable of clause c_j, and as a shorthand write c_j as a set of variables in the clause c_j. In some later proofs this is done, and I do not think it is addressed.
ANSWER: done (a few lines after Theorem 1)
16) Page 6, Section 4, line 224: Add the phrase "weakly positive" right after "said to be". These words are missing.
ANSWER: removed the sentence (it was just a typo). The definition has been provided a few lines above
17) Page 7, Section 4.1, lines 245-249: I strongly suggesting rephrasing this paragraph so the construction is described in terms of "adding" nodes and "adding" arcs to the signed network G. When I first read this, I thought properties were about to be described, not the construction itself.
ANSWER: done
18) Page 7, Section 4.1, line 261: Add the word "The" before "next" so it reads "The next lemma proves..."
ANSWER: done
19) Page 7, Proof of Lemma 1, line 254: Please elaborate on this more. I had a hard time being convinced of this. Can you give a bit more details here?
ANSWER: done
20) Page 11, Section 5.1, line 384: I suggest reminding the reader what "initiators" is, it has been a little while since this term has been used.
ANSWER: done
21) Page 11, Section 5.1, line 394: Maybe use \qedhere or \hfill to put end of proof symbol on the right side. Check the rest of your proofs for the same.
ANSWER: we just used the editor format. But you are right: it looks ugly. So, we are following your suggestion
22) Page 12, Section 5.1, line 427: What is Claim 3? Is this Theorem 3? Lemma 3?
ANSWER: Claim 3 is stated just above the line you refer to. By "claim" we intended a fact we claim to be true, something that has to be remarked (and requires a very short reasoning to get convinced it is true) but that is not enough, let's say, important to be called theorem or lemma. That's the meaning of our Claim 1, Claim 2, Claim 3. But you are right: maybe the word "claim" is misleading. We have replaced it by "fact".
23) Page 12, Section 5.1, lines 426 and 429: Here n is used a couple times. Should it not be |V|?
ANSWER: right. Replaced
24) Page 14, Section 6, lines 486-488: Maybe remove the word "have" from these lines, these are things presented in the paper.
ANSWER: removed
25)Page 14, Section 6, line 488: What is "first"? We are at the end of the paper, you can state this restriction if a "first set of restrictions" is relevant to describe here.
ANSWER: removed
26) Page 14, Section 6, line 491-492: When "graph" is written, can this be written more specifically here?
ANSWER: sorry, we do not understand. The topologies are specified (generalized trees and generalized chains). Maybe, you refer to the form of the sentence and to the use of "-": we have modified it.
27) Page 14, Section 6, line 497: I suggest removing the word "still" from near the start of the sentence.
ANSWER: done
REMARKS FROM REVIEWER 3:
1) P.2 L.33 In the Independent Cascade Model (and other models appearing
in the introduction), is the input graph directed or undirected?
Please clarify it. If the graph is undirected, the notation {u, v}
should be used instead of (u, v) to represent an arc.
ANSWER: generally speaking, the graphs are directed, and we have added "directed" according to your suggestion
2) P.5 Sec 3 in the definition of N^-_in
N^-_in(u) = {v \in H: ...} -> N^-_in(u) = {v \in V: ...}
ANSWER: done
3) P.5 Sec 3 in the definition of N^-_out
N^-_out(u) = {v \in V(u,v) \in A^-} -> N^-_out(u) = {v \in V: (u,v) \in A^-}
ANSWER: done
4) P.6 L.203
all node- and arc-weights equals 1. -> all node- and arc-weights equal 1.
ANSWER: done
5) P.6 L.224
A set of clauses...is said to be
The sentence is incomplete.
ANSWER: the incomplete sentence has been removed (it was a typo, the definition being provided a few lines above)
6) P.6 L.227
at least one clause in f is positive
Please define "a clause is positive" around here.
(Currently, the definition is located in L.239.)
ANSWER: the definition of positive and negative clauses has been moved in the right place
7) P.6 L.227
Indeed, if some clause in f contains a single positive variable t
You mean
"Indeed, if some clause in f consists of just a single positive variable t" ?
if some clause in f contains a single negative variable
is also.
ANSWER: you are right. Done
8) P.8 L.284
since I_0 \subset V_f
->
since I_0 \subseteq V_X
ANSWER: done
9) P.9 between L.321 and L.322
Let u be any node let \pi^+(u) be the set
->
Let u be any node and let \pi^+(u) be the set
ANSWER: done
10) P.9 between L.321 and L.322
Q contains a set Q_u of clauses
Probably, the authors mean "Q is the union of sets Q_u of
clauses associated to u"
ANSWER: the latex formatting made it unclear. Modified
11) P.10 L.345
d^+(I_0, u) <= d^+(I_0, h)
->
d^+(I_0, u) <= d^+(I_0, v)
ANSWER: done
12) P.10 L.370
What is G^H ?
ANSWER: it is G^+. Corrected
13) P.11 L.408
only if there exist z
->
only if there exists z
ANSWER: actually, we have changed this part of the proof since it was quite redundant
14) P.12 L.412
S(\epsilon(v_j) = V
->
S(\epsilon(v_j)) = V
ANSWER: done
15) P.13 between L.448 and L.449
w_G(S(E_1) -> w_G(S(E_1))
w_G(S(E_1) -> w_G(S(E_2)) (The index is incorrect and the right parenthesis misses.)
w_G(S(E_1 \cup E_2) -> w_G(S(E_1 \cup E_2))
ANSWER: done
Reviewer 2 Report
Review for algorithms-689180
Authors: Miriam Di Ianni, Giovanna Varricchio
The paper considers a variety of problems surrounding target set selection in graphs called signed networks. The authors present several reductions, leading to inapproximability results, and special cases are considered where their observations lead to algorithmic results and algorithms with polynomial running times in the worst case.
I think the problem itself is interesting and has not been studied yet. Personally, I am not as knowledgeable of this area of optimization problems, but it reminds me of another problem called the Graph Burning problem (this might even be worth mentioning somewhere in the paper if it is not already) with more facets attached to it, where the model studied in your paper can be used to model many more things such as social networks and diffusion. If applicable, perhaps citing one of these papers may be relevant for the paper; though, I do not think it is necessary as the literature review of this paper is generally comprehensive. I think the results are interesting; however, my opinion might not be representative of an expert opinion for these types of optimization problems.
Generally I did not run into any major problems with this paper. I thought the proofs were generally well presented, and I did not run into any major issues; though, in some places some more explanation would help (as mentioned in the specific comments). I mostly have comments to improve presentation and if there are any issues, my comments might help show those issues to exist. Please consider the specific comments below.
Note: I did not have time to carefully examine Section 5.1, however it mostly read fine.
Specific comments:
-Page 1, Abstract, line 1: Maybe be a bit more specific here, some information does not spread via social networks; perhaps just rephrasing this sentence may make it read better.
-Page 1, Abstract, line 4: perhaps, add the word "problem" after "Target Set Selection". This is done later in the abstract.
-Page 1, Abstract, line 7: What is "it" in this sentence? Maybe being a bit more specific here will be helpful for the reader.
-Page 1, Abstract, line 10: What is "first" in this context? I've read the paper and I do not think writing this helps the reader understand the restriction; maybe rephrasing this sentence may help. It does not need to be specific on what the restriction is, but "first" might not mean much here.
-Page 1, Section 1, line 14: Maybe capitalize the "p" in "processes"; perhaps also add a hyphen between "complex" and "networks".
-Page 2, Section 1, line 33: Should the word "directed" be added between "edge-weighted" and "graph"? Somewhere here this should be clarified as later references may change in how they are interpreted by if the graph is a directed graph or not.
-Page 2, Section 1, line 66: 16 steps? If so, write "16 steps" in the parentheses. I was unsure here.
-Page 3, Section 1, line 116: Add a comma between u and v so that it reads "if $v$ is activated by $u$, $v$ gets...".
-Page 4, Section 2, line 159 and line 163: Several times you use the phrases like Min-Horn Deletion-complete and Set Cover-hard. This is not as common of phrasing, so I suggest you write out once exactly what you mean by this; I'd recommend when you first write Min Horn Deletion-complete.
-Page 4, Section 2, line 162: Maybe change 1-target set -> 1-Target Set; sometimes k-Target Set is used, other times k-target set is used. I suggest picking one and sticking with it. For example, this is done again on line 169 and also on Page 5 on line 184.
-Page 5, Section 3: Maybe add a figure that shows a signed graph, and is used to illustrate the definitions for each +-version; in particular, the definition of $d^+(V',u)$ and $\pi^+(u)$.
-Page 5, Section 3, line 191 (7 lines down): When you write "length" and "shortest", I suggest being more careful here. For example, in the number of arcs? Either way, being more precise here for both "maximum length" and "shortest positive path" will help the reader.
-Page 5, Section 3, line 192 (one line down): Several times in the paper you use the set of natural numbers, I suggest either changing these to $\mathbb{Z}^+$ or being more explicit on the set of natural numbers (and which interpretation of them you are using). You use the natural numbers denoted this way many times, and some people might assume you are including 0 in, while others might not.
-Page 6, Section 3, line 200: I recommend explicitly defining the size of a target set (the one we want as small as possible).
-Page 6, Section 4, line 220: Here you might want to add a sentence stating that at times you will say x_i \in c_j to mean x_i is a variable of clause c_j, and as a shorthand write c_j as a set of variables in the clause c_j. In some later proofs this is done, and I do not think it is addressed.
-Page 6, Section 4, line 224: Add the phrase "weakly positive" right after "said to be". These words are missing.
-Page 7, Section 4.1, lines 245-249: I strongly suggesting rephrasing this paragraph so the construction is described in terms of "adding" nodes and "adding" arcs to the signed network G. When I first read this, I thought properties were about to be described, not the construction itself.
-Page 7, Section 4.1, line 261: Add the word "The" before "next" so it reads "The next lemma proves..."
-Page 7, Proof of Lemma 1, line 254: Please elaborate on this more. I had a hard time being convinced of this. Can you give a bit more details here?
-Page 11, Section 5.1, line 384: I suggest reminding the reader what "initiators" is, it has been a little while since this term has been used.
-Page 11, Section 5.1, line 394: Maybe use \qedhere or \hfill to put end of proof symbol on the right side. Check the rest of your proofs for the same.
-Page 12, Section 5.1, line 427: What is Claim 3? Is this Theorem 3? Lemma 3?
-Page 12, Section 5.1, lines 426 and 429: Here n is used a couple times. Should it not be |V|?
-Page 14, Section 6, lines 486-488: Maybe remove the word "have" from these lines, these are things presented in the paper.
-Page 14, Section 6, line 488: What is "first"? We are at the end of the paper, you can state this restriction if a "first set of restrictions" is relevant to describe here.
-Page 14, Section 6, line 491-492: When "graph" is written, can this be written more specifically here?
-Page 14, Section 6, line 497: I suggest removing the word "still" from near the start of the sentence.
Author Response
First of all, we want to thank you for your careful (and fast!) work on our paper, and for your comments that do helped us to improve it.
According to all reviewers' comments, all our changes are minor ones (typos, os some sintax mistake, or a required figure, or something like that) but the following two slightly more than minor ones:
1) pag. 7, line 243: we are showing that we can consider weakly positive boolean formulas in which at least one clause is positive *without loss of generality*. In the submitted version, the proof of this fact was longer than needed, since it is sufficient to notice that instances in which all clauses are negative do trivially admit a solution of size 0 (and, hence, they are solvable in polynomial-time)
2) pag. 12, three lines beneath line 416: the proof that S(E_i) \subseteq S(E_{i+1} is a direct consequence of E_i \subseteq E_{i+1}, as outlined in this new version. In the submitted version the proof was long and involved.
In what follows, please find your remarks and our answers. For the sake of completeness, in order you have access to all our changes, after that we also include the remarks from the other reviewers as well as our answers.
YOUR REMARKS:
0) "it reminds me of another problem called the Graph Burning problem (this might even be worth mentioning somewhere in the paper if it is not already) with more facets attached to it, where the model studied in your paper can be used to model many more things such as social networks and diffusion. If applicable, perhaps citing one of these papers may be relevant for the paper; though, I do not think it is necessary as the literature review of this paper is generally comprehensive. I think the results are interesting; however, my opinion might not be representative of an expert opinion for these types of optimization problems."
ANSWER: Yes, there are several analogies between diffusion processes and epidemics spreading (or what you call Graph burning). In this last setting our "negative influence" translates in agents contrasting the disease / fire spreading. Actually, there is a huge amount of literature concerning the so-called Firefighter problem. Our model (and our results) can be read also within the firefighter setting, as well as most of the literature related to diffusion processes is also suitable to describe epidemiological processes. We have decided to describe our model in the diffusion processes setting because it looks more natural. And adding a comparison to the literature about firefighting would have resulted really too long. So, we have decided to follow the same approach of the diffusion literature we have cited: disregarding the other side.
1) Page 1, Abstract, line 1: Maybe be a bit more specific here, some information does not spread via social networks; perhaps just rephrasing this sentence may make it read better.
ANSWER: done
2) Page 1, Abstract, line 4: perhaps, add the word "problem" after "Target Set Selection". This is done later in the abstract.
ANSWER: actually, the sentence is "A widely studied problem in the area of viral marketing is the Target Set Selection" and adding "problem" at the end would sound repetitive. Maybe we could say "Target Set Selection one", but it does not sound so fine
3) Page 1, Abstract, line 7: What is "it" in this sentence? Maybe being a bit more specific here will be helpful for the reader.
ANSWER: done
4) Page 1, Abstract, line 10: What is "first" in this context? I've read the paper and I do not think writing this helps the reader understand the restriction; maybe rephrasing this sentence may help. It does not need to be specific on what the restriction is, but "first" might not mean much here.
ANSWER: removed
5) Page 1, Section 1, line 14: Maybe capitalize the "p" in "processes"; perhaps also add a hyphen between "complex" and "networks".
ANSWER: the word "Diffusion" is capitalized just because it is the first word in the sentence If it were in the middle of a sentence, we would have written "diffusion processes". It is not the name of a topic, it is just a noun with its adjective. Writing "complex-networks" is more unusual than "complex networks": see, for instance, the entry in Wikipedia, or the name of the related conference (Complex networks, indeed), or even, more remarkably, Easley-Kleinberg's book https://www.cs.cornell.edu/home/kleinber/networks-book/networks-book.pdf and the literature cited therein.
6) Page 2, Section 1, line 33: Should the word "directed" be added between "edge-weighted" and "graph"? Somewhere here this should be clarified as later references may change in how they are interpreted by if the graph is a directed graph or not.
ANSWER: generally speaking, the graphs are directed, and we have added "directed" according to your suggestion. Since an undirected graph is a particular directed graph (in which arcs (u,v) and (v,u) both exists or both does not exist), this does not change the way the (rather informal) discussion of the results in the literature presented in the introduction can be interpreted.
7) Page 2, Section 1, line 66: 16 steps? If so, write "16 steps" in the parentheses. I was unsure here.
ANSWER: done
8) Page 3, Section 1, line 116: Add a comma between u and v so that it reads "if $v$ is activated by $u$, $v$ gets...".
ANSWER: comma added (although the sentence has been modified according to a different reviewer)
9) Page 4, Section 2, line 159 and line 163: Several times you use the phrases like Min-Horn Deletion-complete and Set Cover-hard. This is not as common of phrasing, so I suggest you write out once exactly what you mean by this; I'd recommend when you first write Min Horn Deletion-complete.
ANSWER: added a footnote and clarified the sentence
10) Page 4, Section 2, line 162: Maybe change 1-target set -> 1-Target Set; sometimes k-Target Set is used, other times k-target set is used. I suggest picking one and sticking with it. For example, this is done again on line 169 and also on Page 5 on line 184.
ANSWER: done
11) Page 5, Section 3: Maybe add a figure that shows a signed graph, and is used to illustrate the definitions for each +-version; in particular, the definition of $d^+(V',u)$ and $\pi^+(u)$.
ANSWER: done
12) Page 5, Section 3, line 191 (7 lines down): When you write "length" and "shortest", I suggest being more careful here. For example, in the number of arcs? Either way, being more precise here for both "maximum length" and "shortest positive path" will help the reader.
ANSWER: in graph theory, a directed/undirected path is defined as a sequence of arcs/edges (see, just to be easy, Wikipedia, but there is plenty of documents in the network) and, hence, the length of a path is defined as the number of arcs/edges composing it. The Shortest Path problem is defined accordingly (and on this definition Dijkstra's algorithm is based). Since it is the standard definition, we just refer to it (being more specific would sound redundant).
13) Page 5, Section 3, line 192 (one line down): Several times in the paper you use the set of natural numbers, I suggest either changing these to $\mathbb{Z}^+$ or being more explicit on the set of natural numbers (and which interpretation of them you are using). You use the natural numbers denoted this way many times, and some people might assume you are including 0 in, while others might not.
ANSWER: done
14) Page 6, Section 3, line 200: I recommend explicitly defining the size of a target set (the one we want as small as possible).
ANSWER: well, the size of a set is, trivially, its cardinality. We have added (a line above) the sentence "Needless to say, the size of a target set $I_0 \subseteq V$ is $|I_0|$."
15) Page 6, Section 4, line 220: Here you might want to add a sentence stating that at times you will say x_i \in c_j to mean x_i is a variable of clause c_j, and as a shorthand write c_j as a set of variables in the clause c_j. In some later proofs this is done, and I do not think it is addressed.
ANSWER: done (a few lines after Theorem 1)
16) Page 6, Section 4, line 224: Add the phrase "weakly positive" right after "said to be". These words are missing.
ANSWER: removed the sentence (it was just a typo). The definition has been provided a few lines above
17) Page 7, Section 4.1, lines 245-249: I strongly suggesting rephrasing this paragraph so the construction is described in terms of "adding" nodes and "adding" arcs to the signed network G. When I first read this, I thought properties were about to be described, not the construction itself.
ANSWER: done
18) Page 7, Section 4.1, line 261: Add the word "The" before "next" so it reads "The next lemma proves..."
ANSWER: done
19) Page 7, Proof of Lemma 1, line 254: Please elaborate on this more. I had a hard time being convinced of this. Can you give a bit more details here?
ANSWER: done
20) Page 11, Section 5.1, line 384: I suggest reminding the reader what "initiators" is, it has been a little while since this term has been used.
ANSWER: done
21) Page 11, Section 5.1, line 394: Maybe use \qedhere or \hfill to put end of proof symbol on the right side. Check the rest of your proofs for the same.
ANSWER: we just used the editor format. But you are right: it looks ugly. So, we are following your suggestion
22) Page 12, Section 5.1, line 427: What is Claim 3? Is this Theorem 3? Lemma 3?
ANSWER: Claim 3 is stated just above the line you refer to. By "claim" we intended a fact we claim to be true, something that has to be remarked (and requires a very short reasoning to get convinced it is true) but that is not enough, let's say, important to be called theorem or lemma. That's the meaning of our Claim 1, Claim 2, Claim 3. But you are right: maybe the word "claim" is misleading. We have replaced it by "fact".
23) Page 12, Section 5.1, lines 426 and 429: Here n is used a couple times. Should it not be |V|?
ANSWER: right. Replaced
24) Page 14, Section 6, lines 486-488: Maybe remove the word "have" from these lines, these are things presented in the paper.
ANSWER: removed
25)Page 14, Section 6, line 488: What is "first"? We are at the end of the paper, you can state this restriction if a "first set of restrictions" is relevant to describe here.
ANSWER: removed
26) Page 14, Section 6, line 491-492: When "graph" is written, can this be written more specifically here?
ANSWER: sorry, we do not understand. The topologies are specified (generalized trees and generalized chains). Maybe, you refer to the form of the sentence and to the use of "-": we have modified it.
27) Page 14, Section 6, line 497: I suggest removing the word "still" from near the start of the sentence.
ANSWER: done
REMARKS FROM REVIEWER 1:
1) ``Epidemiology''->``epidemiology'' (global comment)
ANSWER: done
2) The first sentence of Sec.~1 is hard to read: modified first sentence in Sec. 1
ANSWER: done. The original sentence "Diffusion processes are a widely investigated complex networks related topic aiming at analyzing how local interactions among nearby nodes may lead to the diffusion, possibly over the whole network, of some feature (opinion/information/disease) starting from a limited set of \emph{initiator} (or \emph{seed}) nodes. " has been modified to "The study of diffusion processes is a widely investigated topic in the complex networks setting. It aims
at analyzing how local interactions among nearby nodes may lead to the diffusion, possibly over the whole network, of some feature (opinion/information/disease) starting from a limited set of \emph{initiator} (or \emph{seed}) nodes. "
3) ``their state''->``their states''
ANSWER: done
4) ``each currently unaware neighbor v of u changes its state to informed with probability equals to''->grammer error?
ANSWER: replaced "equals" by "equal" - we hope this was the problem
5) Line~56, p.~2: I think the footnote should appear after the period? (global comment)
ANSWER: the two footnotes have been removed and their content has been moved within the text
6) P.~2: ``it has been shown a polynomial-time algorithm ...''->weird
ANSWER: the sentence has been modified to "... a polynomial-time algorithm spreading a feature over a set of required size within a constant number of time steps in graphs of bounded clique-width has been proposed"
7) P.~2: ``... as to latency-bounded diffusion processes''->weird
ANSWER: sentence changed to "in this paper, diffusion processes with an available fixed number of time steps within which informing all the nodes will be called {\em latency-bounded} diffusion processes"
8) Lines~116--117, p.~3: Too many occurrences of ``with''
ANSWER: sentence changed to "if $v$ is activated by $u$ then the probability that $v$ gets informed with the same orientation of $u$ is $\varphi_{u,v}$, and the probability that $v$ gets informed with the opposite orientation is $1-\varphi_{u,v}$"
9) Lines~118--119, p.~3: ``positive active''->``positively active'' and similarly for negative activation
ANSWER: done
10) Lines~143--144, p.~4: ``people decisions''->``people's decisions''
ANSWER: done
11) Line~192, p.~5: ``smaller than $\theta(u)$''->``less than $\theta(u)$''
ANSWER: done
12) Eq.~(1), p.~6: Use $\cdots$ instead of $\ldots$
ANSWER: done (in Eq. 1 and in the next formula)
13) Line~203, p.~6: ``All ... weights equals ...''->``All ... weights equal ...''
ANSWER: done
14) Line~368, p.~10: Use $\cdots$ to skip operations done over many objects and $\ldots$ for plain enumeration (so if we take $\land$'s, use $\cdots$)
ANSWER: done, tried to fix anywhere (thanks!)
15)Line~427, p.~12: ``an $(1+\log n)$-approximate solution''->``a $(1+\log n)$-approximate solution''
ANSWER: done
16) Line~492, p.~14: Using the hyphen as em dash looks weird (global comment)
ANSWER: fixed the two points in Sec. Conclusions and open problems (not found elsewhere)
REMARKS FROM REVIEWER 3:
1) P.2 L.33 In the Independent Cascade Model (and other models appearing
in the introduction), is the input graph directed or undirected?
Please clarify it. If the graph is undirected, the notation {u, v}
should be used instead of (u, v) to represent an arc.
ANSWER: generally speaking, the graphs are directed, and we have added "directed" according to your suggestion
2) P.5 Sec 3 in the definition of N^-_in
N^-_in(u) = {v \in H: ...} -> N^-_in(u) = {v \in V: ...}
ANSWER: done
3) P.5 Sec 3 in the definition of N^-_out
N^-_out(u) = {v \in V(u,v) \in A^-} -> N^-_out(u) = {v \in V: (u,v) \in A^-}
ANSWER: done
4) P.6 L.203
all node- and arc-weights equals 1. -> all node- and arc-weights equal 1.
ANSWER: done
5) P.6 L.224
A set of clauses...is said to be
The sentence is incomplete.
ANSWER: the incomplete sentence has been removed (it was a typo, the definition being provided a few lines above)
6) P.6 L.227
at least one clause in f is positive
Please define "a clause is positive" around here.
(Currently, the definition is located in L.239.)
ANSWER: the definition of positive and negative clauses has been moved in the right place
7) P.6 L.227
Indeed, if some clause in f contains a single positive variable t
You mean
"Indeed, if some clause in f consists of just a single positive variable t" ?
if some clause in f contains a single negative variable
is also.
ANSWER: you are right. Done
8) P.8 L.284
since I_0 \subset V_f
->
since I_0 \subseteq V_X
ANSWER: done
9) P.9 between L.321 and L.322
Let u be any node let \pi^+(u) be the set
->
Let u be any node and let \pi^+(u) be the set
ANSWER: done
10) P.9 between L.321 and L.322
Q contains a set Q_u of clauses
Probably, the authors mean "Q is the union of sets Q_u of
clauses associated to u"
ANSWER: the latex formatting made it unclear. Modified
11) P.10 L.345
d^+(I_0, u) <= d^+(I_0, h)
->
d^+(I_0, u) <= d^+(I_0, v)
ANSWER: done
12) P.10 L.370
What is G^H ?
ANSWER: it is G^+. Corrected
13) P.11 L.408
only if there exist z
->
only if there exists z
ANSWER: actually, we have changed this part of the proof since it was quite redundant
14) P.12 L.412
S(\epsilon(v_j) = V
->
S(\epsilon(v_j)) = V
ANSWER: done
15) P.13 between L.448 and L.449
w_G(S(E_1) -> w_G(S(E_1))
w_G(S(E_1) -> w_G(S(E_2)) (The index is incorrect and the right parenthesis misses.)
w_G(S(E_1 \cup E_2) -> w_G(S(E_1 \cup E_2))
ANSWER: done
Reviewer 3 Report
This paper considers the Unweighted (or weighted) Signed Target Set
Selection problem (USTSS or STSS). In this problem, a signed directed
graph G = (V, A^+ \cup A^-), a vertex set I_0, called the target set,
a node-weight function \theta: V -> N and a edge-weight function w:
A^+ \cup A^- -> N are given as an input. Each vertex has one of two
status: unaware or informed. First, all the vertices in I_0 are
informed and the rest vertices are unaware. For each discrete time
step, each unaware vertex u becomes informed if the cumulative weight
of its in-friends (indicated by an arc in A^+) is at least \theta(u)
while the cumulative weight of its in-enemies (indicated by an arc in
A^-) is smaller than \theta(u). The goal of USTSS problem is to
minimize the size of the target set such that all the vertices become
informed at the end of the process. The t-USTSS problem asks for
computing minimum size target sets such that the diffusion process
finishes until t time step.
The contributions of the paper are theoretical analysis of the
problem. Specifically, the authors provide inapproxibility of the
2-USTSS (and USTSS) and approximation algorithms for the 1-USTSS.
I think this paper makes a nice contribution to the research field. I
would recommend to acceptance after the minor issues in the following
comments are fixed.
Minor comments:
P.2 L.33 In the Independent Cascade Model (and other models appearing
in the introduction), is the input graph directed or undirected?
Please clarify it. If the graph is undirected, the notation {u, v}
should be used instead of (u, v) to represent an arc.
P.5 Sec 3 in the definition of N^-_in
N^-_in(u) = {v \in H: ...}
->
N^-_in(u) = {v \in V: ...}
P.5 Sec 3 in the definition of N^-_out
N^-_out(u) = {v \in V(u,v) \in A^-}
->
N^-_out(u) = {v \in V: (u,v) \in A^-}
P.6 L.203
all node- and arc-weights equals 1.
->
all node- and arc-weights equal 1.
P.6 L.224
A set of clauses...is said to be
The sentence is incomplete.
P.6 L.227
at least one clause in f is positive
Please define "a clause is positive" around here.
(Currently, the definition is located in L.239.)
P.6 L.227
Indeed, if some clause in f contains a single positive variable t
You mean
"Indeed, if some clause in f consists of just a single positive variable t" ?
L.231
if some clause in f contains a single negative variable
is also.
P.8 L.284
since I_0 \subset V_f
->
since I_0 \subseteq V_X
P.9 between L.321 and L.322
Let u be any node let \pi^+(u) be the set
->
Let u be any node and let \pi^+(u) be the set
P.9 between L.321 and L.322
Q contains a set Q_u of clauses
Probably, the authors mean "Q is the union of sets Q_u of
clauses associated to u"
P.10 L.345
d^+(I_0, u) <= d^+(I_0, h)
->
d^+(I_0, u) <= d^+(I_0, v)
P.10 L.370
What is G^H ?
P.11 L.408
only if there exist z
->
only if there exists z
P.12 L.412
S(\epsilon(v_j) = V
->
S(\epsilon(v_j)) = V
P.13 between L.448 and L.449
w_G(S(E_1)
->
w_G(S(E_1))
w_G(S(E_1)
->
w_G(S(E_2))
(The index is incorrect and the right parenthesis misses.)
w_G(S(E_1 \cup E_2)
->
w_G(S(E_1 \cup E_2))
Author Response
First of all, we want to thank you for your careful (and fast!) work on our paper, and for your comments that do helped us to improve it.
According to all reviewers' comments, all our changes are minor ones (typos, os some sintax mistake, or a required figure, or something like that) but the following two slightly more than minor ones:
1) pag. 7, line 243: we are showing that we can consider weakly positive boolean formulas in which at least one clause is positive *without loss of generality*. In the submitted version, the proof of this fact was longer than needed, since it is sufficient to notice that instances in which all clauses are negative do trivially admit a solution of size 0 (and, hence, they are solvable in polynomial-time)
2) pag. 12, three lines beneath line 416: the proof that S(E_i) \subseteq S(E_{i+1} is a direct consequence of E_i \subseteq E_{i+1}, as outlined in this new version. In the submitted version the proof was long and involved.
In what follows, please find your remarks and our answers. For the sake of completeness, in order you have access to all our changes, after that we also include the remarks from the other reviewers as well as our answers.
YOUR REMARKS:
1) P.2 L.33 In the Independent Cascade Model (and other models appearing
in the introduction), is the input graph directed or undirected?
Please clarify it. If the graph is undirected, the notation {u, v}
should be used instead of (u, v) to represent an arc.
ANSWER: generally speaking, the graphs are directed, and we have added "directed" according to your suggestion
2) P.5 Sec 3 in the definition of N^-_in
N^-_in(u) = {v \in H: ...} -> N^-_in(u) = {v \in V: ...}
ANSWER: done
3) P.5 Sec 3 in the definition of N^-_out
N^-_out(u) = {v \in V(u,v) \in A^-} -> N^-_out(u) = {v \in V: (u,v) \in A^-}
ANSWER: done
4) P.6 L.203
all node- and arc-weights equals 1. -> all node- and arc-weights equal 1.
ANSWER: done
5) P.6 L.224
A set of clauses...is said to be
The sentence is incomplete.
ANSWER: the incomplete sentence has been removed (it was a typo, the definition being provided a few lines above)
6) P.6 L.227
at least one clause in f is positive
Please define "a clause is positive" around here.
(Currently, the definition is located in L.239.)
ANSWER: the definition of positive and negative clauses has been moved in the right place
7) P.6 L.227
Indeed, if some clause in f contains a single positive variable t
You mean
"Indeed, if some clause in f consists of just a single positive variable t" ?
if some clause in f contains a single negative variable
is also.
ANSWER: you are right. Done
8) P.8 L.284
since I_0 \subset V_f
->
since I_0 \subseteq V_X
ANSWER: done
9) P.9 between L.321 and L.322
Let u be any node let \pi^+(u) be the set
->
Let u be any node and let \pi^+(u) be the set
ANSWER: done
10) P.9 between L.321 and L.322
Q contains a set Q_u of clauses
Probably, the authors mean "Q is the union of sets Q_u of
clauses associated to u"
ANSWER: the latex formatting made it unclear. Modified
11) P.10 L.345
d^+(I_0, u) <= d^+(I_0, h)
->
d^+(I_0, u) <= d^+(I_0, v)
ANSWER: done
12) P.10 L.370
What is G^H ?
ANSWER: it is G^+. Corrected
13) P.11 L.408
only if there exist z
->
only if there exists z
ANSWER: actually, we have changed this part of the proof since it was quite redundant
14) P.12 L.412
S(\epsilon(v_j) = V
->
S(\epsilon(v_j)) = V
ANSWER: done
15) P.13 between L.448 and L.449
w_G(S(E_1) -> w_G(S(E_1))
w_G(S(E_1) -> w_G(S(E_2)) (The index is incorrect and the right parenthesis misses.)
w_G(S(E_1 \cup E_2) -> w_G(S(E_1 \cup E_2))
ANSWER: done
REMARKS FROM REVIEWER 1:
1) ``Epidemiology''->``epidemiology'' (global comment)
ANSWER: done
2) The first sentence of Sec.~1 is hard to read: modified first sentence in Sec. 1
ANSWER: done. The original sentence "Diffusion processes are a widely investigated complex networks related topic aiming at analyzing how local interactions among nearby nodes may lead to the diffusion, possibly over the whole network, of some feature (opinion/information/disease) starting from a limited set of \emph{initiator} (or \emph{seed}) nodes. " has been modified to "The study of diffusion processes is a widely investigated topic in the complex networks setting. It aims
at analyzing how local interactions among nearby nodes may lead to the diffusion, possibly over the whole network, of some feature (opinion/information/disease) starting from a limited set of \emph{initiator} (or \emph{seed}) nodes. "
3) ``their state''->``their states''
ANSWER: done
4) ``each currently unaware neighbor v of u changes its state to informed with probability equals to''->grammer error?
ANSWER: replaced "equals" by "equal" - we hope this was the problem
5) Line~56, p.~2: I think the footnote should appear after the period? (global comment)
ANSWER: the two footnotes have been removed and their content has been moved within the text
6) P.~2: ``it has been shown a polynomial-time algorithm ...''->weird
ANSWER: the sentence has been modified to "... a polynomial-time algorithm spreading a feature over a set of required size within a constant number of time steps in graphs of bounded clique-width has been proposed"
7) P.~2: ``... as to latency-bounded diffusion processes''->weird
ANSWER: sentence changed to "in this paper, diffusion processes with an available fixed number of time steps within which informing all the nodes will be called {\em latency-bounded} diffusion processes"
8) Lines~116--117, p.~3: Too many occurrences of ``with''
ANSWER: sentence changed to "if $v$ is activated by $u$ then the probability that $v$ gets informed with the same orientation of $u$ is $\varphi_{u,v}$, and the probability that $v$ gets informed with the opposite orientation is $1-\varphi_{u,v}$"
9) Lines~118--119, p.~3: ``positive active''->``positively active'' and similarly for negative activation
ANSWER: done
10) Lines~143--144, p.~4: ``people decisions''->``people's decisions''
ANSWER: done
11) Line~192, p.~5: ``smaller than $\theta(u)$''->``less than $\theta(u)$''
ANSWER: done
12) Eq.~(1), p.~6: Use $\cdots$ instead of $\ldots$
ANSWER: done (in Eq. 1 and in the next formula)
13) Line~203, p.~6: ``All ... weights equals ...''->``All ... weights equal ...''
ANSWER: done
14) Line~368, p.~10: Use $\cdots$ to skip operations done over many objects and $\ldots$ for plain enumeration (so if we take $\land$'s, use $\cdots$)
ANSWER: done, tried to fix anywhere (thanks!)
15)Line~427, p.~12: ``an $(1+\log n)$-approximate solution''->``a $(1+\log n)$-approximate solution''
ANSWER: done
16) Line~492, p.~14: Using the hyphen as em dash looks weird (global comment)
ANSWER: fixed the two points in Sec. Conclusions and open problems (not found elsewhere)
REMARKS FROM REVIEWER 2:
0) "it reminds me of another problem called the Graph Burning problem (this might even be worth mentioning somewhere in the paper if it is not already) with more facets attached to it, where the model studied in your paper can be used to model many more things such as social networks and diffusion. If applicable, perhaps citing one of these papers may be relevant for the paper; though, I do not think it is necessary as the literature review of this paper is generally comprehensive. I think the results are interesting; however, my opinion might not be representative of an expert opinion for these types of optimization problems."
ANSWER: Yes, there are several analogies between diffusion processes and epidemics spreading (or what you call Graph burning). In this last setting our "negative influence" translates in agents contrasting the disease / fire spreading. Actually, there is a huge amount of literature concerning the so-called Firefighter problem. Our model (and our results) can be read also within the firefighter setting, as well as most of the literature related to diffusion processes is also suitable to describe epidemiological processes. We have decided to describe our model in the diffusion processes setting because it looks more natural. And adding a comparison to the literature about firefighting would have resulted really too long. So, we have decided to follow the same approach of the diffusion literature we have cited: disregarding the other side.
1) Page 1, Abstract, line 1: Maybe be a bit more specific here, some information does not spread via social networks; perhaps just rephrasing this sentence may make it read better.
ANSWER: done
2) Page 1, Abstract, line 4: perhaps, add the word "problem" after "Target Set Selection". This is done later in the abstract.
ANSWER: actually, the sentence is "A widely studied problem in the area of viral marketing is the Target Set Selection" and adding "problem" at the end would sound repetitive. Maybe we could say "Target Set Selection one", but it does not sound so fine
3) Page 1, Abstract, line 7: What is "it" in this sentence? Maybe being a bit more specific here will be helpful for the reader.
ANSWER: done
4) Page 1, Abstract, line 10: What is "first" in this context? I've read the paper and I do not think writing this helps the reader understand the restriction; maybe rephrasing this sentence may help. It does not need to be specific on what the restriction is, but "first" might not mean much here.
ANSWER: removed
5) Page 1, Section 1, line 14: Maybe capitalize the "p" in "processes"; perhaps also add a hyphen between "complex" and "networks".
ANSWER: the word "Diffusion" is capitalized just because it is the first word in the sentence If it were in the middle of a sentence, we would have written "diffusion processes". It is not the name of a topic, it is just a noun with its adjective. Writing "complex-networks" is more unusual than "complex networks": see, for instance, the entry in Wikipedia, or the name of the related conference (Complex networks, indeed), or even, more remarkably, Easley-Kleinberg's book https://www.cs.cornell.edu/home/kleinber/networks-book/networks-book.pdf and the literature cited therein.
6) Page 2, Section 1, line 33: Should the word "directed" be added between "edge-weighted" and "graph"? Somewhere here this should be clarified as later references may change in how they are interpreted by if the graph is a directed graph or not.
ANSWER: generally speaking, the graphs are directed, and we have added "directed" according to your suggestion. Since an undirected graph is a particular directed graph (in which arcs (u,v) and (v,u) both exists or both does not exist), this does not change the way the (rather informal) discussion of the results in the literature presented in the introduction can be interpreted.
7) Page 2, Section 1, line 66: 16 steps? If so, write "16 steps" in the parentheses. I was unsure here.
ANSWER: done
8) Page 3, Section 1, line 116: Add a comma between u and v so that it reads "if $v$ is activated by $u$, $v$ gets...".
ANSWER: comma added (although the sentence has been modified according to a different reviewer)
9) Page 4, Section 2, line 159 and line 163: Several times you use the phrases like Min-Horn Deletion-complete and Set Cover-hard. This is not as common of phrasing, so I suggest you write out once exactly what you mean by this; I'd recommend when you first write Min Horn Deletion-complete.
ANSWER: added a footnote and clarified the sentence
10) Page 4, Section 2, line 162: Maybe change 1-target set -> 1-Target Set; sometimes k-Target Set is used, other times k-target set is used. I suggest picking one and sticking with it. For example, this is done again on line 169 and also on Page 5 on line 184.
ANSWER: done
11) Page 5, Section 3: Maybe add a figure that shows a signed graph, and is used to illustrate the definitions for each +-version; in particular, the definition of $d^+(V',u)$ and $\pi^+(u)$.
ANSWER: done
12) Page 5, Section 3, line 191 (7 lines down): When you write "length" and "shortest", I suggest being more careful here. For example, in the number of arcs? Either way, being more precise here for both "maximum length" and "shortest positive path" will help the reader.
ANSWER: in graph theory, a directed/undirected path is defined as a sequence of arcs/edges (see, just to be easy, Wikipedia, but there is plenty of documents in the network) and, hence, the length of a path is defined as the number of arcs/edges composing it. The Shortest Path problem is defined accordingly (and on this definition Dijkstra's algorithm is based). Since it is the standard definition, we just refer to it (being more specific would sound redundant).
13) Page 5, Section 3, line 192 (one line down): Several times in the paper you use the set of natural numbers, I suggest either changing these to $\mathbb{Z}^+$ or being more explicit on the set of natural numbers (and which interpretation of them you are using). You use the natural numbers denoted this way many times, and some people might assume you are including 0 in, while others might not.
ANSWER: done
14) Page 6, Section 3, line 200: I recommend explicitly defining the size of a target set (the one we want as small as possible).
ANSWER: well, the size of a set is, trivially, its cardinality. We have added (a line above) the sentence "Needless to say, the size of a target set $I_0 \subseteq V$ is $|I_0|$."
15) Page 6, Section 4, line 220: Here you might want to add a sentence stating that at times you will say x_i \in c_j to mean x_i is a variable of clause c_j, and as a shorthand write c_j as a set of variables in the clause c_j. In some later proofs this is done, and I do not think it is addressed.
ANSWER: done (a few lines after Theorem 1)
16) Page 6, Section 4, line 224: Add the phrase "weakly positive" right after "said to be". These words are missing.
ANSWER: removed the sentence (it was just a typo). The definition has been provided a few lines above
17) Page 7, Section 4.1, lines 245-249: I strongly suggesting rephrasing this paragraph so the construction is described in terms of "adding" nodes and "adding" arcs to the signed network G. When I first read this, I thought properties were about to be described, not the construction itself.
ANSWER: done
18) Page 7, Section 4.1, line 261: Add the word "The" before "next" so it reads "The next lemma proves..."
ANSWER: done
19) Page 7, Proof of Lemma 1, line 254: Please elaborate on this more. I had a hard time being convinced of this. Can you give a bit more details here?
ANSWER: done
20) Page 11, Section 5.1, line 384: I suggest reminding the reader what "initiators" is, it has been a little while since this term has been used.
ANSWER: done
21) Page 11, Section 5.1, line 394: Maybe use \qedhere or \hfill to put end of proof symbol on the right side. Check the rest of your proofs for the same.
ANSWER: we just used the editor format. But you are right: it looks ugly. So, we are following your suggestion
22) Page 12, Section 5.1, line 427: What is Claim 3? Is this Theorem 3? Lemma 3?
ANSWER: Claim 3 is stated just above the line you refer to. By "claim" we intended a fact we claim to be true, something that has to be remarked (and requires a very short reasoning to get convinced it is true) but that is not enough, let's say, important to be called theorem or lemma. That's the meaning of our Claim 1, Claim 2, Claim 3. But you are right: maybe the word "claim" is misleading. We have replaced it by "fact".
23) Page 12, Section 5.1, lines 426 and 429: Here n is used a couple times. Should it not be |V|?
ANSWER: right. Replaced
24) Page 14, Section 6, lines 486-488: Maybe remove the word "have" from these lines, these are things presented in the paper.
ANSWER: removed
25)Page 14, Section 6, line 488: What is "first"? We are at the end of the paper, you can state this restriction if a "first set of restrictions" is relevant to describe here.
ANSWER: removed
26) Page 14, Section 6, line 491-492: When "graph" is written, can this be written more specifically here?
ANSWER: sorry, we do not understand. The topologies are specified (generalized trees and generalized chains). Maybe, you refer to the form of the sentence and to the use of "-": we have modified it.
27) Page 14, Section 6, line 497: I suggest removing the word "still" from near the start of the sentence.
ANSWER: done
Round 2
Reviewer 1 Report
The paper is OK. Just double check for typos, etc. For example:
Line~159, p.~4: ``that that''->repeated?
Line~167, p.~4: ``Set Cover-hard''->Elsewhere you use smallcaps for computational problems. Be consistent.
Line~471, p.~14: ``an''->``a''
Author Response
Thank you very much for your comments. We have corrected these typos.
Reviewer 2 Report
I think the authors have addressed the main comments on the paper. I only have one small comment:
Page 7, line 231: Should there be "to be" between ""said" and "weakly positive"? It seems like something is missing there.
Author Response
Thank you very much for your comments. We have corrected these typos.
Reviewer 3 Report
All the issues I pointed out are fixed.
Author Response
Thank you very much.
