Thanks for the comments and please kindly check our further clarifications (continued)
Additionally, we would like to clarify that:
1. **The motivation of this work is to highlight the issue of “NP-hard masking” in neural solvers and enhance neural solvers with the ability to handle complex constraints**, as comprehensively demonstrated in our main paper (Tables 1, 2, and 3). Thus, outperforming LKH3 is not our primary objective or key contribution. Instead, our proposed framework has significantly reduced the infeasibility rate and substantially improved solution quality for both autoregressive and non-autoregressive neural solvers. These advancements establish our framework as the new state-of-the-art in the domain of neural solvers.
2. **Most existing NCO solvers [2-8] use LKH3 with the default settings from [1]** (including maximum trials, number of runs, and parallelism settings) to estimate the optimality gap and provide a runtime reference. The primary focus is on understanding the performance gap between neural solvers [10, 11] and traditional solvers, rather than strictly outperforming it.
3. We acknowledge that **neural solvers have yet to fully replace traditional solvers like LKH3, but rather catching up with it step by step**. We believe that we provide insights for complex constraints handling for NCO, which is a step towards practical applications. Also, we would like to emphasize that our PIP(-D) is a **generic framework**, making our research orthogonal to the development of new, powerful neural solvers that have the potential to fully outperform LKH in future works.
We appreciate the reviewer for helping our work to be more rigorous. The new results will be incorporated into the revised manuscript, along with a deeper discussion of these points. Due to the rebuttal limit, we will plot the progress curves (like Figure R2 and R3 in the attached PDF) based on the results of all the new tables for a more intuitive illustration. Also, **our code has already been forwarded to the AC**. We hope our discussions are now comprehensive. Please let us know if you have any remaining concerns.
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References:
[1] Attention, learn to solve routing problems!. ICLR, 2018.
[2] Reinforcement learning for solving the vehicle routing problem. NeurIPS, 2018.
[3] POMO: Policy optimization with multiple optima for reinforcement learning. NeurIPS, 2022.
[4] Simulation-guided beam search for neural combinatorial optimization. NeurIPS, 2022.
[5] Learning to search feasible and infeasible regions of routing problems with flexible neural k-opt. NeurIPS, 2023.
[6] BQ-NCO: Bisimulation quotienting for generalizable neural combinatorial optimization. NeurIPS, 2023.
[7] Neural combinatorial optimization with heavy decoder: Toward large scale generalization. NeurIPS, 2023.
[8] MVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts. ICLM, 2024.
[9] Machine learning for combinatorial optimization: a methodological tour d’horizon. European Journal of Operational Research, 2021.
[10] Graph-based diffusion solvers for combinatorial optimization. NeurIPS, 2023.
[11] MOCO: A Learnable Meta Optimizer for Combinatorial Optimization. arXiv preprint, 2024.
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Note for Tables R7 & R8: For a fair comparison, we kept the POMO*+PIP(-D) models (with inference time denoted as $T_{PIP}$) unchanged and allowed LKH3 to search for a similar duration as $T_{PIP}$ to showcase the performance of POMO*+PIP(-D)+LKH3 (i.e., the total inference time is $2 \times T_{PIP}$). LKH3 was also given the same time limit for consistency.