StruDiCO: Structured Denoising Diffusion with Gradient-free Inference-stage Boosting for Memory and Time Efficient Combinatorial Optimization

Dear Reviewer 2ZPS,

Thanks for your insightful comments and for acknowledging novelty, model design, and empirical performance. Below we respond to the specific comments.

Q1&W2: Some recent diffusion-based solvers have been overlooked in the comparison, such as DISCO: Efficient Diffusion Solver for Large-Scale Combinatorial Optimization Problems, which also explores improving both solution quality and inference speed for CO problems. A technical discussion of the differences from these works, along with public performance comparisons, would further strengthen the completeness and positioning of this work. Whether the inference cost of StruDiCO increases significantly with problem size, such as TSP-10000. While this does not affect the rating, clarifying this point would be helpful.

Thanks for providing the related work. DISCO improves inference efficiency and solution quality on large-scale CO problems (e.g., TSP, MIS) via residue-constrained denoising and multi-modal graph search. We have now incorporated a discussion of DISCO in the related work section.

We have extended our model training to TSP-10,000 and compared it with diffusion-based baselines, including DIFUSCO and DISCO, and a divide-and-conquer (DC) approach, GLOP [1].

It is important to note that as the problem size increases (e.g., 100K nodes), the primary bottleneck during inference is not the diffusion (or consistency) model itself. Rather, the major computational overhead arises from the greedy decoding procedure, where edges or nodes with the highest confidence are sequentially inserted into a partial solution until a feasible one is constructed. This step can become prohibitively time-consuming in large-scale instances. We believe that the scalability of StruDiCO could be further improved by combining its backbone generative model with a more efficient decoding strategy or a divide-and-conquer approach. This presents a promising direction for future research.

We will include these large-scale results and comparisons with DISCO in the final version to strengthen the contribution.

W1: The reviewer is curious about the contribution of the two key diffusion design components, Variable-Absorption and Constrained Consistency Sampling, to the overall performance. Conducting a thorough ablation study would help clarify their individual effects.

We provide a detailed ablation study to isolate the contribution of each component in our method. The analysis focuses on three parts:

1. The role of Variable-Absorption (VA) mechanism,
2. The effect of Constrained Consistency Sampling (CCS) in the main inference process, and
3. The contribution of the Objective-Aware Refinement module applied on the initial solution solved by StruDiCO with $T_s=5$.

Specifically, we compare several variants, including:

• Uniform CM: the uniform cosistency model (CM) proposed in Fast T2T;
• VA CM: replacing the uniform CM with the Variable-Absorption CM;
• +CCS: where low-confidence variables are masked;
• Vanilla Refinement: which refines the initial solution without gradient-free objective guidance (OG).

Table: Ablation Study (Gap, % ↓)

As shown, replacing the uniform noise (used in DIFUSCO, T2T, and FastT2T) with the Variable-Absorption (VA) mechanism alone leads to a performance degradation. This is because VA introduces a monotonic absorption process, where variables are only deactivated (i.e., 1 → 0) during the forward diffusion. While this preserves structural continuity at each timestep, it results in overly conservative trajectories when used alone—limiting exploration and diversity in the reverse process. In contrast, the Uniform Consistency Model (CM) allows symmetric noise injection (both 0 ↔ 1), which encourages broader exploration and increases the chance of reaching high-quality solutions.

The key point we emphasize is that VA was not designed to be effective in isolation. Rather, its role is to create structurally consistent and interpretable intermediate states, serving as a reliable foundation for the reverse process when combined with CCS. Once CCS is applied, which selectively activates only high-confidence variables at each step, the VA framework’s structure-preserving property becomes crucial: it ensures that CCS operates over a low-noise, semantically meaningful space. Notably, while CCS also improves performance when applied to the Uniform CM baseline, the gain is significantly amplified when combined with VA. This confirms that these two components are complementary and jointly responsible for the observed performance gains.

Building on these improved initial solutions, the introduction of Vanilla Refinement further boosts solution quality. On top of that, the addition of Objective Guidance (OG) steers the refinement process by prioritizing variables with both high model confidence and low surrogate cost, in a fully gradient-free manner with negligible runtime overhead.

In summary, this ablation study highlights the modular and complementary design of our method:

• VA + CCS collaboratively ensure structural reliability, enabling better solution space exploration and improved outcomes;
• OG promotes objective alignment effectively and efficiently, resulting in better solutions.

We will include this extended analysis and clarification in the final version to better support the empirical strength of our approach.

Supplementary Experiments on a New Problem: Capacitated Vehicle Routing Problem

At the end of this rebuttal, we would like to present an additional experiment on the Capacitated Vehicle Routing Problem (CVRP), which we believe provides further strong empirical support for our methodology. Due to space limitations, detailed results are provided in our response to Reviewer RWGz, Q3.

To the best of our knowledge, StruDiCO is the first heatmap-based model to surpass sequence-based methods on CVRP, highlighting the potential of structure-guided global prediction in solving combinatorial optimization problems with complex constraints.

References

[1] GLOP: Learning Global Partition and Local Construction for Solving Large-scale Routing Problems in Real-time, AAAI, 2024.

We hope this response could help ease your concern and wish to receive your further feedback soon.

Best regards,

The Authors

Dear Reviewer 2ZPS,

Thank you again for your thoughtful review and for highlighting both the strengths and the potential of our proposed method.

As the NeurIPS 2025 author-reviewer discussion phase has been extended, we would like to kindly follow up in case you had any additional thoughts after reading our rebuttal and supplementary results.

We noticed that the rating has been updated (though hidden from us, as per policy), and we sincerely hope that our detailed responses have addressed your concerns. If there are any remaining issues you'd like to further discuss, or if you would like to briefly comment on whether our clarifications were satisfactory, we would greatly appreciate your feedback.

In particular, we responded in detail to your comments regarding:

• A comprehensive ablation study analyzing the contributions of Variable Absorption (VA), Constrained Consistency Sampling (CCS), and Objective-Guided Refinement (OG);

• Scalability on TSP-10,000, and comparisons with DISCO and GLOP;

• The practical applicability of our method to more constrained problems like CVRP, demonstrating generalization beyond TSP and MIS.

Please don’t hesitate to let us know if any further clarification would help you finalize your assessment.

Thank you again for your time and consideration.

Best regards,

The Authors

Dear Reviewer XM73,

Thanks for your insightful comments and for acknowledging model design and empirical performance. Below we respond to the specific comments.

Q1: Clarifying the individual contributions of Variable-Absorption and Constrained Consistency Sampling via ablation.

We provide a detailed ablation study to isolate the contribution of each component in our method. The analysis focuses on three parts:

1. The role of Variable-Absorption (VA) mechanism,
2. The effect of Constrained Consistency Sampling (CCS) in the main inference process, and
3. The contribution of the Objective-Aware Refinement module applied on the initial solution solved by StruDiCO with $T_s=5$.

Specifically, we compare several variants, including:

• Uniform CM: the uniform cosistency model (CM) proposed in Fast T2T;
• VA CM: replacing the uniform CM with the Variable-Absorption CM;
• +CCS: where low-confidence variables are masked;
• Vanilla Refinement: which refines the initial solution without gradient-free objective guidance (OG).

Table: Ablation Study (Gap, % ↓)

As shown, replacing the uniform noise (used in DIFUSCO, T2T, and FastT2T) with the Variable-Absorption (VA) mechanism alone leads to a performance degradation. This is because VA introduces a monotonic absorption process, where variables are only deactivated (i.e., 1 → 0) during the forward diffusion. While this preserves structural continuity at each timestep, it results in overly conservative trajectories when used alone—limiting exploration and diversity in the reverse process. In contrast, the Uniform Consistency Model (CM) allows symmetric noise injection (both 0 ↔ 1), which encourages broader exploration and increases the chance of reaching high-quality solutions.

The key point we emphasize is that VA was not designed to be effective in isolation. Rather, its role is to create structurally consistent and interpretable intermediate states, serving as a reliable foundation for the reverse process when combined with CCS. Once CCS is applied, which selectively activates only high-confidence variables at each step, the VA framework’s structure-preserving property becomes crucial: it ensures that CCS operates over a low-noise, semantically meaningful space. Notably, while CCS also improves performance when applied to the Uniform CM baseline, the gain is significantly amplified when combined with VA. This confirms that these two components are complementary and jointly responsible for the observed performance gains.

Building on these improved initial solutions, the introduction of Vanilla Refinement further boosts solution quality. On top of that, the addition of Objective Guidance (OG) steers the refinement process by prioritizing variables with both high model confidence and low surrogate cost, in a fully gradient-free manner with negligible runtime overhead.

In summary, this ablation study highlights the modular and complementary design of our method:

• VA + CCS collaboratively ensure structural reliability, enabling better solution space exploration and improved outcomes;
• OG promotes objective alignment effectively and efficiently, resulting in better solutions.

We will include this extended analysis and clarification in the final version to better support the empirical strength of our approach.

Q2: The decoding strategy is guided by a multiplicative score between model confidence and task-specific penalties. This formulation seems simplistic and may bias the search toward either confidence or task cost.

Thank you for your constructive comment. To better explore the trade-off between model confidence and task-specific penalties, we extended our original formulation. Specifically, we tested a generalized decoding score defined as: $confidence / (\epsilon + penalty)$, where ε serves as a balancing term. Notably, our original multiplicative form can be viewed as a special case: ε = 0 for TSP and ε = 1 for MIS.

In our new experiments, we found that the TSP task is insensitive to ε due to its relatively smooth objective landscape. However, for MIS, tuning ε led to noticeable improvements. As shown in the table below, setting ε = 10⁻² for RB graphs and ε = 10⁻³ for ER graphs yielded better results than the original ε = 1 setting.

Table: Effect of ε on MIS Gap (% ↓)

These results suggest that our decoding formulation is not inherently biased, but rather flexible and tunable to different CO tasks. We will include these findings and the general formulation in the final version.

Q3: Explaining the fast convergence of gradient-free refinement.

Thank you for the insightful question. Unlike classical local search methods such as 2-opt, which typically apply a single local perturbation per step and require thousands of iterations to converge, our refinement mechanism performs a batch update: at each step, it simultaneously perturbs all low-confidence variables using variable-absorption noising, followed by selection based on objective-aware decoding. This parallel refinement strategy significantly accelerates convergence.

To support this, we present the following results on TSP-500, showing the trade-off between performance and computation time over different refinement steps:

Table: Refinement Step Efficiency (TSP-500)

As shown, the refinement process leads to steady improvements over steps. Notably, the most significant quality gains are achieved within the first 3–5 steps, making this range a good trade-off point between performance and efficiency. Although further steps continue to improve the solution, the marginal gain diminishes, while the computational cost increases linearly.

Q4: Scalability of StruDiCO and comparison with large-scale CO solvers (e.g., DISCO).

To evaluate the scalability of our method, we conducted additional experiments on the large-scale TSP-10,000 dataset. We compare StruDiCO with diffusion-based baselines, including DIFUSCO and DISCO, and a divide-and-conquer (DC) approach, GLOP [1].

Table: Results on TSP-10000

It is important to note that as the problem size increases (e.g., 100K nodes), the primary bottleneck during inference is not the diffusion (or consistency) model itself. Rather, the major computational overhead arises from the greedy decoding procedure, where edges or nodes with the highest confidence are sequentially inserted into a partial solution until a feasible one is constructed. This step can become prohibitively time-consuming in large-scale instances. We believe that the scalability of StruDiCO could be further improved by combining its backbone generative model with a more efficient decoding strategy or a divide-and-conquer approach. This presents a promising direction for future research.

We will include these large-scale results and comparisons with DISCO in the final version to strengthen the contribution.

Q5: Code release.

We appreciate your suggestion. We are committed to releasing our full codebase and pretrained models upon publication. The objective of this work is to provide reference and contributions for the ML4CO community. Open-sourcing the code is a fundamental obligation we undertake.

Supplementary Experiments on a New Problem: Capacitated Vehicle Routing Problem

At the end of this rebuttal, we present an additional experiment on the CVRP, which we believe would provide additional strong empirical support for our methodology. Due to space limitations, please refer to our response to Reviewer RWGz, Q3, for the results.

References

[1] GLOP: Learning Global Partition and Local Construction for Solving Large-scale Routing Problems in Real-time, AAAI, 2024.

We hope this response could help address your concerns, and we are more than happy to address any further concerns you may have.

Best regards,

The Authors

Dear Reviewer RWGz,

Thanks for the thorough review and valuable comments. Below we respond to your specific comments.

Q1: The improvements in TSP and MIS look minimal. Can you elaborate on why the performance gap is small, and under what settings the proposed method provides more substantial benefits?

We respectfully disagree with the view that the improvements on TSP and MIS are minimal. Across various settings, StruDiCO achieves average relative performance gains of 19.6% (TSP) and 38.1% (MIS) over the strongest diffusion-based baselines (e.g., Fast T2T). While the absolute reductions in optimality gap may appear numerically small, it is important to recognize that these improvements are achieved in an extremely low-error regime, where prior methods were already within 1% of optimality. Improving further under such conditions is inherently more challenging and thus more meaningful.

Moreover, StruDiCO delivers substantial improvements in efficiency. In particular, StruDiCO introduces a gradient-free refinement mechanism that significantly accelerates inference compared to existing gradient-based refinement methods (e.g., T2T, Fast T2T). Specifically, we observe up to 70% reduction in GPU memory usage and a 3.5$\times$ speedup in inference time, as shown in Tables 2–5 and Fig. 2 of the main paper. These results demonstrate that StruDiCO advances both solution quality and computational efficiency, offering a practical and scalable alternative to gradient-based refinement.

Beyond performance, StruDiCO also enhances interpretability by maintaining structurally valid intermediate states throughout the diffusion process. This structure-preserving design facilitates step-wise understanding of the solution construction, enabling deeper analysis of combinatorial patterns—an aspect particularly valuable for operations research and scientific decision-making applications.

Please kindly note that reviewers Z5NJ, XM73, and 2ZPS all acknowledge the empirical results of our method.

As for the settings under which our method offers more substantial benefits, we observe that the improvements are especially pronounced in larger-scale and structurally complex problem instances—such as TSP-1000 and MIS-ER (see Fig. 2). In these challenging scenarios, the advantages of structure-preserving diffusion and gradient-free objective-aware refinement become more evident, leading to larger gains in both solution quality and efficiency.

Q2: Can you provide a more detailed ablation that showcases the contribution of the gradient-free refinement? How much does each component in StruDiCO contribute to the final performance improvement?

Thank you for the thoughtful question. We provide a detailed ablation study to isolate the contribution of each component in our method. The analysis focuses on two parts:

1. The effect of Constrained Consistency Sampling (CCS) in the main inference process, and
2. The contribution of the gradient-free objective-aware refinement module applied on the initial solution solved by StruDiCO with $T_s=5$.

Specifically, we compare several variants, including:

• VA only: using the Variable-Absorption diffusion with greedy decoding;
• +CCS: where low-confidence variables are masked;
• Vanilla Refinement: which refines the initial solution without gradient-free objective guidance (OG).

Table: Ablation Study (Gap, % ↓)

From the results, we first observe that incorporating Constrained Consistency Sampling (CCS) significantly enhances the performance of Variable Absorption (VA), yielding up to a 51% reduction in optimality gap (e.g., MIS-ER: 23.461 → 11.450). This improvement arises because CCS prevents low-confidence variables from being prematurely activated, resulting in more stable and reliable inference trajectories.

Building on these improved initial solutions, the introduction of Vanilla Refinement further boosts solution quality. On top of that, the addition of Objective Guidance (OG) steers the refinement process by prioritizing variables with both high model confidence and low surrogate cost—in a fully gradient-free manner with negligible runtime overhead. For example: TSP-500 improves from 0.381 → 0.319 ; MIS-ER improves from 7.611 → 5.579

In summary, this ablation study highlights the modular and complementary design of our method:

• VA + CCS collaboratively ensure structural reliability, enabling better solution space exploration and improved outcomes;
• OG promotes objective alignment effectively and efficiently, resulting in better solutions.

Owing to space limitations, this response primarily focuses on clarifying the contribution to performance improvements. For a more comprehensive ablation analysis, please refer to our response to Reviewer XM73, Q1. In addition, Fig. 2 in the main paper provides a detailed comparison with gradient-based refinement methods, demonstrating that our gradient-free refinement consistently outperforms them in both solution quality and computational efficiency.

Q3: Could StruDiCO be generalized well to more CO problems?

In fact, StruDiCO can handle a wide range of constrained combinatorial optimization (CO) problems. Leveraging the expressive capacity of generative models, our approach learns to approximate the underlying constraint structure, producing heatmaps that are already close to feasible. We then apply lightweight heuristic post-processing (e.g., greedy decoding) to convert these outputs into strictly feasible solutions. This paradigm enables general applicability to arbitrary constraints and aligns with the inference strategies adopted in prior representative works such as Diffusco, T2T, and Fast T2T.

To further demonstrate the generality of StruDiCO beyond classical edge-selection (TSP) and node-selection (MIS) tasks, we extend it to the Capacitated Vehicle Routing Problem (CVRP)—a more challenging CO task that involves both global routing structures and hard feasibility constraints, such as customer demands and vehicle capacities. The refinement procedure for CVRP follows the same design as that for TSP: greedy decoding is performed based on the learned heatmap, with the objective guided by a distance-based penalty term. We evaluate StruDiCO on CVRP-50, CVRP-100, and CVRP-200 under standard benchmark settings [1]. To further improve both solution feasibility and cost, we incorporate a local search heuristic (Classic-LS [2]) during inference.

To the best of our knowledge, StruDiCO is the first heatmap-based model that surpasses sequence-based models on VRPs.

W2: Gradient-free approach is not critical in tested examples: the tested tasks TSP and MIS do not present known issues with gradient use, so it’s unclear when this feature would be critical.

We respectfully clarify that although TSP and MIS are inherently discrete and non-differentiable problems, recent diffusion-based solvers, such as T2T and FastT2T, still rely on gradient-based refinement over relaxed continuous representations to improve solution quality. In contrast, our method achieves superior performance without any gradient computation, demonstrating that a gradient-free refinement framework can be both effective and efficient. This not only avoids the complexity and overhead associated with backpropagation through relaxed spaces but also enhances the method’s applicability to a broader range of combinatorial problems where gradient-based refinement may be infeasible or suboptimal.

References

[1] RL4CO: An Extensive Reinforcement Learning for Combinatorial Optimization Benchmark, 2024

[2] A Simple and Effective Evolutionary Algorithm for the Vehicle Routing Problem, 2004

[3] A Hybrid Genetic Algorithm for Multidepot and Periodic Vehicle Routing Problems, 2012

We hope our point-by-point clarificaitons and supplementary experiments have satisfactorily addressed your concerns. We remain fully committed to involving further discussions with you towards an elevated evaluation of our work.

Best regards,

The Authors

Dear Reviewer RWGz,

Thank you again for your review and thoughtful comments.

As the NeurIPS 2025 program chairs have extended the author-reviewer discussion phase, we would like to kindly follow up to ensure you’ve had a chance to revisit our detailed rebuttal and supplementary results. We would greatly appreciate your thoughts or clarifications if any part of our response warrants further discussion.

In particular, we have carefully addressed the key concerns you raised:

• On the perceived marginal improvements: We respectfully clarified that StruDiCO achieves average relative gains of 19.6% (TSP) and 38.1% (MIS) over strong diffusion-based baselines (e.g., Fast T2T), in an already low-error regime. We also showed that these gains are more pronounced on larger-scale or structurally complex instances, such as TSP-1000 and MIS-ER.

• On the contribution of gradient-free refinement: We provided a detailed ablation study, which demonstrated the complementary effects of Variable Absorption (VA), Constrained Consistency Sampling (CCS), and Objective-Guided Refinement (OG), as well as the efficiency and effectiveness of our fully gradient-free approach.

• On generalizability beyond TSP and MIS: We extended StruDiCO to the Capacitated Vehicle Routing Problem (CVRP), showing that it not only handles more complex constraints but also outperforms sequence-based models on standard CVRP benchmarks.

• On the necessity of gradient-free refinement: We clarified that while TSP and MIS are discrete tasks, recent diffusion solvers (e.g., T2T, Fast T2T) still rely on gradient-based refinement over continuous relaxations. In contrast, StruDiCO offers a practical and scalable alternative without requiring gradient computations.

We sincerely value your feedback and would be grateful if you could let us know whether these responses help clarify your concerns, or if there are additional points you’d like to discuss. Your input would be invaluable in helping us further improve and present our work as clearly and rigorously as possible.

Thank you again for your time and effort in reviewing our submission.

Best regards,

The Authors

Dear Reviewer Z5NJ,

Thanks for your recognition and valuable questions.

Q1: Clarification on why the proposed method achieves higher accuracy than other diffusion-based methods.

To better explain the performance gain of our method over existing diffusion-based approaches (e.g., those using uniform diffusion), we performed an ablation study to isolate the individual contributions of Variable Absorption (VA) and Constrained Consistency Sampling (CCS).

Ablation Study (Gap, %↓):

As shown, replacing the uniform noise (used in DIFUSCO, T2T, and FastT2T) with the Variable-Absorption (VA) mechanism alone leads to a performance degradation. This is because VA introduces a monotonic absorption process, where variables are only deactivated (i.e., 1 → 0) during the forward diffusion. While this preserves structural continuity at each timestep, it results in overly conservative trajectories when used alone, limiting exploration and diversity in the reverse process. In contrast, the Uniform Consistency Model (CM) allows symmetric noise injection (both 0 ↔ 1), which encourages broader exploration and increases the chance of reaching high-quality solutions.

The key point we emphasize is that VA was not designed to be effective in isolation. Rather, its role is to create structurally consistent and interpretable intermediate states, serving as a reliable foundation for the reverse process when combined with CCS. Once CCS is applied, which selectively activates only high-confidence variables at each step, the VA framework’s structure-preserving property becomes crucial: it ensures that CCS operates over a low-noise, semantically meaningful space. Notably, while CCS also improves performance when applied to the Uniform CM baseline, the gain is significantly amplified when combined with VA. This confirms that these two components are complementary and jointly responsible for the observed performance gains.

This response primarily addresses your concern regarding the impact of diffusion model variations on accuracy. For a more comprehensive ablation analysis, please refer to our response to Reviewer XM73, Q1.

Q2: How much labeled data is enough for your method?

For fairness, we adopt the same training set sizes as those used in DIFUSCO and FastT2T.

W2: The refinement step in StruDiCO is based on heuristics. It is unclear how their method is efficient for the new CO problem.

In fact, StruDiCO can handle a wide range of constrained combinatorial optimization (CO) problems. Leveraging the expressive capacity of generative models, our approach learns to approximate the underlying constraint structure, producing heatmaps that are already close to feasible. We then apply lightweight heuristic post-processing (e.g., greedy decoding) to convert these outputs into strictly feasible solutions. This paradigm enables general applicability to arbitrary constraints and aligns with the inference strategies adopted in prior representative works such as Diffusco, T2T, and Fast T2T.

To further demonstrate the generality of StruDiCO beyond classical edge-selection (TSP) and node-selection (MIS) tasks, we extend it to the Capacitated Vehicle Routing Problem (CVRP)—a more challenging CO task that involves both global routing structures and hard feasibility constraints, such as customer demands and vehicle capacities. The refinement procedure for CVRP follows the same design as that for TSP: greedy decoding is performed based on the learned heatmap, with the objective guided by a distance-based penalty term. We evaluate StruDiCO on CVRP-50, CVRP-100, and CVRP-200 under standard benchmark settings [1]. To further improve both solution feasibility and cost, we incorporate a local search heuristic (Classic-LS [2]) during inference.

To the best of our knowledge, StruDiCO is the first heatmap-based model that surpasses sequence-based models on VRPs.

References

[1] RL4CO: An Extensive Reinforcement Learning for Combinatorial Optimization Benchmark, 2024

[2] A Simple and Effective Evolutionary Algorithm for the Vehicle Routing Problem, 2004

[3] A Hybrid Genetic Algorithm for Multidepot and Periodic Vehicle Routing Problems, 2012

We hope this response could help ease your concern and wish to receive your further feedback soon.

Best regards,

The Authors

Dear Reviewer Z5NJ,

Thank you again for your initial review and thoughtful comments.

As the author-reviewer discussion phase has been extended by 48 hours, we would like to kindly follow up to see if you had a chance to revisit our rebuttal and supplementary results. We have responded in detail to all your concerns, including:

• A thorough ablation analysis addressing why our method outperforms T2T and DIFUSCO in terms of accuracy (Q1);

• Clarification on data requirements, aligned with existing diffusion-based solvers (Q2);

• Additional experiments on CVRP demonstrating the generality of StruDiCO across constrained CO tasks (W2).

Your feedback was instrumental in helping us refine our explanations and highlight the complementary role of Variable Absorption and Constrained Consistency Sampling. We would greatly appreciate it if you could kindly take a moment to let us know whether our responses have addressed your concerns, or if you have any further suggestions or questions.

Thank you once again for your time and support in the review process. We sincerely value your input.

Best regards,

The Authors