We thank the reviewer for the continued discussion. We regret that concerns still remain, however we remain confident that these points can be satisfactorily addressed and do not represent significant issues with our work.
Whilst we hoped our original responses would clarify these points, we are more than happy to address them further. However, **we would kindly ask that the Reviewer clarify specifically why they do not feel our responses were sufficient** such that we can continue to address remaining questions.
### Concern 1
We are uncertain about the Reviewer's specific concerns. Is the question centered on the rationale for having a diverse policy set rather than a single policy when addressing NP-hard CO problems? Or does the Reviewer find the toy example in Figure 1 overly simplistic to illustrate this point? If it is the latter, we are open to substituting the maze with a more intricate routing problem. The overarching argument remains valid, should the Reviewer prefer this change.
However, we wish to highlight that Poppy's robust empirical performance is our principal contribution. Our goal with Figure 1 and the related discussion was to elucidate the intuition behind proposing the Poppy objective in a simplified setting. Even if there are suggestions for alternative motivating examples or potential for deeper theoretical insights, we maintain that Poppy's demonstrated effectiveness justifies our contribution.
### Concern 3
We are confident that our results demonstrate that Poppy produces a set of complementary policies. We'll outline the evidence, and if the reviewer remains unconvinced, we'd value specific feedback. Should our manuscript lack clarity on this topic, we're open to making necessary revisions.
By "a complementary set of policies", we mean:
1. Individual policies in the population are most performant on different sub-distributions of problem instances.
2. As a collective, the policies encompass the entire problem instance distribution, ensuring that the performance of the best policy for any sampled problem instance remains strong.
**Evidence for (1)**
Removing any agent from the population reduces the overall population-level performance, as seen in Fig 4's bottom-right (Appendix D.1). This signifies that every agent is optimal for a specific sub-distribution of problems. It's important to note that these contributions are computed with respect to greedy rollouts of each policy, and so cannot be attributed to stochasticity in the agent performance.
Fig. 3 illustrates the effects of specialization. As the population size grows, the expected performance of a randomly selected agent on a random problem gets worse. However, the performance of the best-suited agent for a random problem is improved. This highlights that when each agent has a narrower target sub-distribution, it learns an even more specialized policy, which is even better (worse) for problem instances in (out) of the target sub-distribution. This is in contrast with a simple policy ensemble, for which the average agent performance would remain the same regardless of the population size.
**Evidence for (2)**
In a head-to-head comparison of Poppy's collective performance against POMO – either sampling a single POMO agent N times, or training an ensemble of N POMO agents – Poppy consistently surpasses both benchmarks at the same cost (Tables 1, 2 and 3). This demonstrates that the set of policies learned by Poppy, as a collective, excels across all problem instances.
### Concern 4
**4.1**: As described in our original response, we agree that scaling to extremely large CO problems is an important challenge but note that it is a distinct research topic. Baseline methods to which Poppy is compared (POMO, EAS, MDAM, LIH, NeuRewriter, NLNS, 2-Opt-DL, CVAE-Opt, DPDP) tackle the same order of problem sizes as our work. This ensures fair comparison and our leading performance in these problem settings (see Tables 1, 2 and 3) is not questioned by any reviewers. If the Reviewer’s belief is that scaling up to TSP1000 is mandatory for all research in this domain, we would respectfully disagree and note this is a standard not applied to prior works.
**4.2.1**: The requested MDAM baseline for KP and JSSP is shown to be significantly outperformed by other RL methods on TSP and CVRP (Tables 1 and 2). Given the absence of published results for MDAM on these problems and the impracticality of re-implementing all existing neural CO methods, we prioritized the strongest baselines: POMO+sampling and POMO+ensembling. If the Reviewer does not agree this is reasonable we would welcome specific inquiries.
**4.2.2**: The requested baseline was included in our original submission, so we believe this issue should be considered resolved.