PONE-D-25-17298The Firefighter problem with dynamic defence costsPLOS ONE

Dear Dr. Hunter,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. The three reviewers suggested accept, minor revision, and reject. I suggest major revision. I suggest addressing all the reviewers' comments. In addition, I think the graphs studied in the paper are very restricted. I suggest significantly improving the paper by considering more general graphs.

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PLOS ONE

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Additional Editor Comments:

The paper introduces a new variant of the classic Firefighter Problem, which models how contagions like rumors or diseases spread through a network. In the traditional problem, a player protects nodes in a graph to stop the spread and save as many as possible. This new version, called the Cost Function Firefighter Problem, adds complexity by making the cost of defending each node vary over time and depend on the graph's burning state. The authors prove that this version is computationally hard, even on tree structures where the original problem is more manageable. However, by formulating the problem in monadic second-order logic, they show it becomes fixed-parameter tractable when considering factors like treewidth, budget, and maximum time step.

In addition to theoretical analysis, the authors carry out empirical studies to compare different heuristic strategies—based on cost, threat, and degree—under varying cost conditions. They test these heuristics on both random and real-world graphs and find that the effectiveness of each approach depends heavily on the underlying graph structure. Notably, degree-based heuristics generally perform worse than strategies that consider the dynamic state of the graph, such as cost or threat.

The three reviewers suggested accept, minor revision, and reject. I suggest major revision. I suggest addressing all the reviewers' comments. In addition, I think the graphs studied in the paper are very restricted. I suggest significantly improving the paper by considering more general graphs.

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: No

Reviewer #2: Yes

Reviewer #3: Partly

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: No

Reviewer #2: N/A

Reviewer #3: Yes

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3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: No

Reviewer #2: Yes

Reviewer #3: Yes

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5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: Report for the manuscript is attached with the system metadata. It can be seen with the system login. I just suggest to prepare a manuscript that shows one track concept and results rather choosing multi directional concepts without technical concepts in the paper.

Reviewer #2: In the paper entitled "The Firefi ghter problem with dynamic defence costs",

the authors have introduced a variant of the Fire ghter problem called The Cost

Function Fire ghter Problem, in which each vertex has a cost to defend that varies

with time and the burning state of the graph. To complement theoretical fi ndings,

they compare performance of cost, threat and degree-based heuristics under various

cost functions. Their findings explain that the relative effectiveness of these heuris-

tics depends heavily on the structure of graph used in the process, but degree-based

heuristics tend to perform worse than state-based strategies. This manuscript has

been presented at an international conference, 2024.

I suggest to accept the manuscript, after authors address the following;

1. How well do these models capture real-world dynamics like delay, uncertainty,

or feedback loops?

2. Has the problem been simulated on real networks (e.g., social, transportation,

biological)?

3. Can the problem be extended to weighted or directed graphs, and how does

that affect solution strategies?

4. What algorithms are commonly used to solve or approximate solutions to the

Fire fighter Problem?

5. What are the best-known approximation ratios for specifi c versions of the

problem?

6. On which graph classes is the Fire fighter Problem solvable in polynomial time?

7. What are the known variants of the Firefi ghter Problem?

8. What are the limitations of this study?

9. Aligns text to both margins evenly.

10. Figure 8, 9, 10, 11 and 12 are not clear.

11. Make sure that all references are cited within the manuscript.

Reviewer #3: Manuscript ID: PONE-D-25-17298

Title: The Firefighter Problem with Dynamic Defence Costs

Strengths

• Excellent structure and clarity throughout.

• Strong theoretical foundation and clear NP-hardness reductions.

• Effective use of MSO logic for parameterized complexity.

• Experimental evaluation is thorough and includes both random and real-world graphs.

________________________________________

Minor Issues & Suggested Revisions

1. Abstract Improvements

• Line: “This problem is computationally hard, but can be solved efficiently...”

o Suggestion: Add a brief example or graph class for which it is efficiently solvable to make the sentence self-contained.

• Line: “We introduce a variant...”

o Suggestion: Consider emphasizing the novelty more clearly — e.g., highlight that it is the first to incorporate time-dependent vertex defence costs.

________________________________________

2. Notation and Definitions

• Definition 2 (Strategy)

o The multi-index notation (e.g., djid_{ji}) is clear, but it may help to include a small illustrative figure for visual clarity.

• Definition 3 (State Mapping)

o It would be beneficial to provide a visual timeline or state diagram for the example.

________________________________________

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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Attachments

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Submitted filename: PONE-D-25-17298.pdf

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Submitted filename: plos1.pdf

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Submitted filename: Review Report-for Plos.docx

The Firefighter problem with dynamic defence costs

PONE-D-25-17298R1

Dear Dr. Hunter,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Kind regards,

Jan Rychtář

Academic Editor

PLOS One

Additional Editor Comments (optional):

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: (No Response)

Reviewer #2: All comments have been addressed

Reviewer #3: All comments have been addressed

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2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: No

Reviewer #2: Yes

Reviewer #3: Yes

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3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: No

Reviewer #2: N/A

Reviewer #3: Yes

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4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: No

Reviewer #2: Yes

Reviewer #3: Yes

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6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: This paper is more theoretical and less mathematical. The Term Graph Theory is used but Graph Theoretic concepts are not studied in the fashion it is required. Most of the results are based on Theoretical Concepts rather Mathematical. Firefighter Problem is a good topic to study, but it is interesting only if studied through mathematical ways. I, therefore, suggest the authors to make the work more mathematical and less theoretical by using mathematical tools in general and graphical tools in structural properties.

Reviewer #2: (No Response)

Reviewer #3: I have carefully reviewed the manuscript in accordance with the journal’s guidelines. The study is original, technically sound, and clearly presented. The methodology is appropriate, and the results are well supported by the analysis. I did not identify any concerns related to dual publication, plagiarism, data integrity, or research and publication ethics. The manuscript does not appear to have been submitted elsewhere, and all references are properly cited.

The authors have adequately addressed relevant literature and clearly stated the contributions of their work. Overall, the paper meets the ethical and academic standards of the journal, and I have no additional ethical concerns to report.

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7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

Reviewer #3: Yes: Sikander Ali

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