Degree-Constrained Minimum Spanning Hierarchies in Graphs

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see my report in the file Molnar.pdf

Comments for author File: Comments.pdf

Comments on the Quality of English Language

I have only a few minor points.

Author Response

Please see the attachment.

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In the present paper, the author investigates the degree-constrained minimum spanning hierarchies (DCMSH) in graphs with non-uniform capacity constraints. It is proposed an ILP-based exact solution, a heuristic algorithm and the existence of approximation ratios is also discussed.

Some recommendations:

1. In the introduction it will be useful to discuss about the variants of the classical MST. We recommend to cite the paper:

P.C. Pop, The generalized minimum spanning tree problem: an overview of formulations, solution procedures and latest advances, European Journal of Operational Research, Vol. 283(1), pp. 1-15, 2020.

which makes a description of various variants of MST.

2. Concerning the computational results it would be interested to check the performance of the proposed approaches on benchmark instances used in the case of the degree-constrained minimum spanning problem.

Author Response

Thank you very much for taking the time to review this manuscript and thank you for your positive opinion.

. Please find the detailed responses below.

Comments 1:  In the introduction it will be useful to discuss about the variants of the classical MST.
We recommend to cite the paper:
P.C. Pop, The generalized minimum spanning tree problem: an overview of formulations, solution procedures
and latest advances, European Journal of Operational Research, Vol. 283(1), pp. 1-15, 2020.
which makes a description of various variants of MST.

Response 1: Thank you for this recommendation. I think that discussing a broad set of variants of the MST
problem opens up a vast area that is not closely related to the specific topic (DCMST - DCMSH) of my article.
Looking at the opening, I found and added  to the introduction a reference closer to the degree constrained
spanning problem: 
Xin Gao, Lifen Jia, Degree-constrained minimum spanning tree problem with uncertain edge weights,
Applied Soft Computing, Volume 56, 2017, Pages 580-588, ISSN 1568-4946

 

Comments 2: Concerning the computational results it would be interested to check the performance of the
proposed approaches on benchmark instances used in the case of the degree-constrained minimum spanning
problem.

Response 2: Thank you for the proposition. Given the date of submission of this special issue of the
journal, and knowing the investigation required for the tests, this analysis cannot be done quickly
and can only be done in a later analysis.

Reviewer 3 Report

Comments and Suggestions for Authors

Please refer to the attached file for the comments and suggestions for authors.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Please refer to the attached file for the comments on the quality of English language. 

Author Response

 

Thank you very much for taking the time to review this manuscript and being very precise in your recommendations. Thank you for your positive opinion on the results of the paper. I followed your recommendations and corrected the problems in the new version of the paper.

Reviewer 4 Report

Comments and Suggestions for Authors

This paper is a progress on the line of the quoted papers [6, 7, 13]. The faced problems are the following two on a connected graph G. Of course there are spanning trees in G; assume that you have a constraint at each vertex v: the degree of v in the spanning tree is not allowed to exceed a fixed bound D(v). Moreover, assume that a cost function is defined on the edge set; the first problem (degree-constrained minimum spanning tree, DCMST) is the search for a spanning tree of minimum total cost among the ones respecting the constraint. But it may happen that no spanning tree of G respects the constraint on all of its vertices; in this case, the problem is relaxed: degree-constrained minimum spanning hierarchy, DCMSH. A hierarchy is a homomorphism from a suitable tree T to G, surjective on the vertex set of G; here the constraint is on the degree of the vertices of T and the cost is summed on all the occurrences of the same edge of G as an image of the homomorphism. This problem is NP-hard.

 

In the first part of the present paper some necessary, sufficient, necessary-and-sufficient conditions are given for the existence of a DCMSH. The remainder is dedicated to the construction of DCMSHs (if any),  to particular classes of graphs, to approximations and bounds for the costs and to a rather rich experimentation.

 

The paper is definitely interesting and clearly written; still, I have a few remarks.

 

l.20

The word "structure" is used here in an informal way, but later (l.118) it appears as a technical term. At line 8 (i.e. in the Abstract) it is specified as a hierarchy, but Property 1 (l.120-121) seems to set a difference between the two concepts. I think that the word "structure" should either 1) be formally defined before Definition 3 and Property 1, or 2) avoided within formal statements as these.

 

p.5, footnote 1

Actually, it should rather be said that remotion makes the number of connected components increase. I understand that there is an implicit condition of connectedness of G, but this should be explicitly said at the beginning of the formal part. My suggestion: add "connected" at line 54, when introducing G.

 

l.178

These paths should not contain either a or b. I believe that the condition on V_1 implies the existence of shortest paths not containing them, but it should be proved. Why not drop the condition of being shortest? It is immediate that paths not containing a and b exist, if no constraint on their length is forced.

 

l.458-459

The form of this statement is not - in my opinion - one of a theorem, but rather of a remark. I suggest either to reformulate it in a more formal way or to label it as Remark.

 

Finally, I have some language-related suggestions, although I am not of English mother-language (see the specific window).

 

In conclusion, I recommend publication after these minor changes.

 

Comments on the Quality of English Language

l.120

Minimum  -->  A minimum   Every minumum

 

l.132

degree constrained spanning tree does  -->  a degree constrained spanning tree does    degree constrained spanning trees do

 

p.5 footnote 2

This footnote should be totally rewritten.

 

l.203

and  -->  such that

 

l.205

"spider" can be intuitively understood but is used here as a formal term. Instead of defining it, I suggest to put it between quotation marks: ``spider''.

 

l.372

eventual  -->  possible    potential

 

l.422

dp  -->  d_p

 

l.434

"If there are a few number of nodes" should be rewritten.

 

l.437

eventually  -->  possibly

 

Author Response

Please find in the attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have revised the paper according to the reviewer's comments and suggestions.

Author Response

Thank you for your time in reviewing and for your helpful comments.

 

Reviewer 4 Report

Comments and Suggestions for Authors

Comments3

l.178

These paths should not contain either a or b. I believe that the condition on V_1 implies the existence of shortest paths not containing them, but it should be proved. Why not drop the condition of being shortest? It is immediate that paths not containing a and b exist, if no constraint on their length is forced.

 

Response3

Thank you for this proposition.

I think the proof is true and simple. The metrical closure can always be built,

Since the set of nodes should be covered, a walk covering the node set is a Hamiltonian path between A

and B in the metrical closure.

I prefer the original proof.

 

Reply:

I think I expressed my doubt in an unclear way. I agree that one must start from a Hamiltonian path in the metric closure. My objection is that you should specify that each edge of the Hamiltonian path ought to be substituted by a walk not having a or b as internal nodes (be it shortest or not). What I require is just to add this condition, not to change the proof.

 

Author Response

 

Comments3 in the first review:

l.178

These paths should not contain either a or b. I believe that the condition on V_1 implies the existence of shortest paths not containing them, but it should be proved. Why not drop the condition of being shortest? It is immediate that paths not containing a and b exist, if no constraint on their length is forced.

 

Response3

Thank you for this proposition.

I think the proof is true and simple. The metrical closure can always be built,

Since the set of nodes should be covered, a walk covering the node set is a Hamiltonian path between A

and B in the metrical closure.

I prefer the original proof.

 

Reply:

I think I expressed my doubt in an unclear way. I agree that one must start from a Hamiltonian path in the metric closure. My objection is that you should specify that each edge of the Hamiltonian path ought to be substituted by a walk not having a or b as internal nodes (be it shortest or not). What I require is just to add this condition, not to change the proof.

Response: Thank you for your comment. I agree, your precision is necessary. The modified proof is as follows:

Suppose there are two nodes a and b with degree bound 1. A complete graph MC  can be constructed in which the edges represent the shortest paths in G that contain neither a nor b as an internal node. These paths exist because V1 does not contain any separators. ...

 

In addition, I slightly modified the conclusion.