Polynomial Time Algorithm for Shortest Paths in Interval Temporal Graphs^†

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Temporal graphs are directed graphs in which the edges have associated temporal information. If any temporal edge from u to v has the form (u,v,t,λ), that means that a communication from u to v may be initiated at time t and the arrival time at v is t+ λ, and the graph is called Contact Sequence Temporal Graph (CSG). In an Interval Temporal Graph (ITG) an edge is represented as (u,v,intvls), where intvls is a vector of contiguous time intervals.

          In this paper the authors present a polynomial time algorithm to find shortest paths in an ITG. The subject is interesting and the algorithm could have various applications. However there are essential parts of the article which are carelessly written. Hence there are unclear statements for the reader and gaps in proving the validity of the algorithm. In order the change a heuristic algorithm into an exact algorithm we propose to solve the following problems:

          Q1. The authors must define exactly the notions and the statements as for example: in p. 6, line 143 must define “valid extension”, p. 7, lines 154-156, explain why algorithm ends in this manner etc.

          Q2. The paper must contains a picture of an ITG which is suitable for applying Algorithm 1. It is necessary to find all P(v,r) and P_new(v,r), r=1,…,k and to obtain a shortest path.

          Q3. Apply also Algorithm 2 to the graph required in Q2.

          Q4. In Section 4 is necessary to give more details concerning the connections with the previous sections (e. g. the graph, the use of the algorithm in this case).

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

The authors proposed a polynomial time algorithm to find the shortest path in interval temporal graphs. The algorithm can be suitable for ITG with high activity factor and large contiguous travel intervals. Some datasets are used to evaluate the performance and efficiency of the method and compare with benchmark method, the results are rather ok. There are some problems could be clarified or revised.

(1) In a temporal path, when a traveler arrives a new node at time t, this node might not active at time t, so the traveler must wait until the node actives. How to consider this waiting time when you find the shortest path? Furthermore, does the wait time consider in the length of the path?

(2) In the experiments, the authors just compare their method with reference 17 which is published in 2016. Some latest methods could be used to compare if possible.

(3) The title of section 4 is not suitable since the content of this section just introduce the dataset. I recommend merge section 4 and 5 with title being "Experimental results". The original section 4 as a subsection of "Experimental results" with subtitle being "Datasets"

Author Response

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Reviewer 3 Report

Comments and Suggestions for Authors

1. This paper has a lot of notations such as G, V, E, etc. A notation table should be added.

2. The figures are kind of weird. I think they should be re-drawn with professional tools, such as draw.io.

3. Please rename algorithms 1 and 2. The names of algorithms 1 and 2 are not specific, which are not recommended.

4. I think CSG and ITG are different types of networks. Then what is the point for comparing the algorithm on ITG with CSG? If they are the same type of network, then why did the authors mention them separately? Please  explain.

5. For Tables 4 and 5, if the running time is based on seconds, the authors could just write down a note saying that. It is not necessary to write down s after all numbers in the tables.

 

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

This new version of the paper improves the previous one.

Author Response

Dear Reviewer,

Thanks for reviewing the updates in the revised manuscript and accepting the changes.

Thanks

Anuj and Sartaj Sahni

Reviewer 2 Report

Comments and Suggestions for Authors

Thank authors revise the manuscript. Some of my comments are considered in the revised manuscript. However, there are still some issues that exist.

(1) Since the waiting time is considered when a traveler vists from source to destination, so, the total traveling time should contain the waiting time. I don't understand why the waiting time is not considered in the shortest temporal path.

(2) In the experiments, some latest reference should be cited and compared, for example, ref[1]. 

[1] Casteigts, A., Himmel, AS., Molter, H. et al. Finding Temporal Paths Under Waiting Time Constraints. Algorithmica 83, 2754–2802 (2021). https://doi.org/10.1007/s00453-021-00831-w

Author Response

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Author Response File: Author Response.docx

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

I have no further comment.