Redesigning the Wheel for Systematic Travelling Salesmen
Round 1
Reviewer 1 Report
The paper proposes a method for solving large TSP instances, i.e., with up to 10 million points (nodes). Specifically, the method applies k-opt heuristic with k from 2 to 6 to improve an initial solution created using an additional heuristic. Several other components have been applied to solve relatively large TSP instances, including Delaunay triangulation for limiting the number of candidate nodes considered when looking for improving moves, a two-level data structure to speed up k-opt moves application, and parallel processing to further shorten the computation time. Probably none of these components is novel in itself; however, I find the novelty in the overall structure of the proposed approach. The author also provides an extensive, detailed, and valuable computational study of the proposed method, comparing it to the well-known LKH heuristic. The results confirm that the algorithm can reach good solutions within the specified 30 minutes time limit.
I have only minor issues with the paper:
- more details should be provided on the programming tools used in the implementation, i.e. programming language, how the parallelism was implemented (OpenMP, MPI, sth else), compiler and compiler switches, etc. I know that this will be available in the source code, however, adding two sentences of explanation in the paper is probably a good idea
- I cannot find in the references a recent paper that considers solving even larger TSP instances with billions of nodes:
Taillard, Éric D. "A linearithmic heuristic for the travelling salesman
problem." European Journal of Operational Research 297.2 (2022): 442-450.
- the author should mention that the performance of the LKH depends strongly on the parameters used, and it could be possible to obtain higher quality solutions if other values of the parameters were used. For example, I have consistently been able to get higher quality (err. <1%) solutions to the C100k.0 instance within a few minutes if the following parameters have been used
SUBGRADIENT = YES, CANDIDATE_SET_TYPE = POPMUSIC, INITIAL_PERIOD = 100
Few minor language issues:
- "insert their points in existing paths" -> "(...) into existing paths"
- "and is reasonable fast" -> "and is reasonably fast"
- "a new pair of point" -> "a new pair of points"
- "provides a god compromise" -> "provides a good compromise"
- "have to be red" -> "have to be read" (but pronounced "red")
- "the elapsed time additional includes" -> "(...) additionally includes"
Author Response
I would like to thank the reviewers for careful reading of the manuscript and providing helpful suggestions for its improvement.
Reviewer1
1.
“more details should be provided on the programming tools used in the implementation, i.e. programming language, how the parallelism was implemented (OpenMP, MPI, sth else), compiler and compiler switches, etc. I know that this will be available in the source code, however, adding two sentences of explanation in the paper is probably a good idea”
=> In subsection 3.1, the sentence “The source code can also be compiled on computers running Windows operation systems.” has been changed to: “The source code (ANSI-C) utilizes the Linux-specific pthread library for parallel processing and has been compiled with gcc and option `-Ofast'. Alternatively, the code can be compiled on computers running Windows operation systems.”
2.
“I cannot find in the references a recent paper that considers solving even larger TSP instances with billions of nodes:
Taillard, Éric D. "A linearithmic heuristic for the travelling salesman problem." European Journal of Operational Research 297.2 (2022): 442-450.”
=> reference has been added in the introductory section (bottom of page 2)
3.
“the author should mention that the performance of the LKH depends strongly on the parameters used, and it could be possible to obtain higher quality solutions if other values of the parameters were used. For example, I have consistently been able to get higher quality (err. <1%) solutions to the C100k.0 instance within a few minutes if the following parameters have been used
SUBGRADIENT = YES, CANDIDATE_SET_TYPE = POPMUSIC, INITIAL_PERIOD = 100”
=> In section 3.8, 2nd paragraph, a new sentence has been added “It has to be mentioned that LKH might result in better tours for smaller instances when using manually optimized settings.”
4.
“Few minor language issues:
- "insert their points in existing paths" -> "(...) into existing paths"
- "and is reasonable fast" -> "and is reasonably fast"
- "a new pair of point" -> "a new pair of points"
- "provides a god compromise" -> "provides a good compromise"
- "have to be red" -> "have to be read" (but pronounced "red")
- "the elapsed time additional includes" -> "(...) additionally includes"
=> These issues and few more typos have been fixed.
Reviewer 2 Report
The paper investigates the usage of k-opt permutations in application to local optimization of very large symmetric two-dimensional TSP instances. The proposed method employs several techniques for accelerating the processing in an intelligent way. The paper is very interesting and well written. However, some minor issues need to be solved:
1. Either "TSP" or "travelling salesman problem" needs to be deleted from the list of keywords (redundant keywords);
2. Lines 27-28: the sentence "Corresponding..." seems not to be included in the context. Please reshape the paragraph.
3. Please increase the linewidth of the lines from Figs. 8-18 to be easily visible.
Author Response
I would like to thank the reviewers for careful reading of the manuscript and providing helpful suggestions for its improvement.
Reviewer 2
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“The paper investigates the usage of k-opt permutations in application to local optimization of very large symmetric two-dimensional TSP instances. The proposed method employs several techniques for accelerating the processing in an intelligent way. The paper is very interesting and well written. However, some minor issues need to be solved:
Either "TSP" or "travelling salesman problem" needs to be deleted from the list of keywords (redundant keywords);”
=> "travelling salesman problem" has been removed
Lines 27-28: the sentence "Corresponding..." seems not to be included in the context. Please reshape the paragraph.
=> the sentence “Corresponding software is freely available for academic use.” Has been changed to “Implementations for exact solvers are freely available for academic use.”
“Please increase the linewidth of the lines from Figs. 8-18 to be easily visible.”
=> line width has been doubled for these plots
Round 2
Reviewer 1 Report
I am satisfied with the changes made.
Reviewer 2 Report
The author has successfully solved all my comments/concerns.
