Rethinking Neural Multi-Objective Combinatorial Optimization via Neat Weight Embedding

• Real-world applications with complex constraints are acknowledged as challenging for this approach. Further exploration into handling such constraints would enhance the paper's practical relevance.

• The paper’s unified training model, WE-CA-U, provides promising generalization across problem sizes, but more discussion on its failure cases (where applicable) would improve understanding of its limitations.

The choice of reference points for evaluating the authors' methods and the baselines raises a concern. In the paper, it is mentioned that CNH [1] bears some similarities to the authors' approach. The CNH paper also evaluates the bi- and tri-objective Traveling Salesman Problem as well as the bi-objective Capacitated Vehicle Routing Problem. Additionally, six out of the twelve baseline methods listed in Table 2 of this paper are also present in Table II of the CNH paper: MOED/D, NSGA-II, MOGLS, DRL-MOA, PMOCO, and PMOCO-Aug. However, the reference points for calculating hypervolume (HV) in this paper differ from those in the CNH paper, making direct comparisons with their results impossible. For reference, please see the table below.

Employing the same reference points as the CNH paper would enable a direct comparison and lend additional credence to the findings if they align with the established results. Without this alignment, and in the absence of publicly available code, it is difficult to verify the results. Addressing this issue would greatly enhance the rigor and transparency of the paper.

I would be glad to reconsider my review if the authors could either provide results using the same reference points as the CNH paper or offer a clear justification for the reference points chosen in this study. This is the concern that has informed my rating for Soundness.

[1] Mingfeng Fan, Yaoxin Wu, Zhiguang Cao, Wen Song, Guillaume Sartoretti, Huan Liu, and Guohua Wu. Conditional neural heuristic for multiobjective vehicle routing problems. IEEE Transactions on Neural Networks and Learning Systems, 2024.

Other concerns about the paper are thus:

1. Figure 2 is intended to show the Pareto fronts, but it doesn’t. For instance, in the figure for KroAB200 (right), the results for CNH-Aug, represented by green circles, show that the left-most point appears to Pareto dominate all other CNH-Aug points. There are more examples across the three figures.
2. Section 3.1 and parts of 3.2 bear significant resemblance to the Section 3.1 and 3.2 of the paper on PMOCO [2]. However, I have overlooked this as they simply discuss the problem formulation.
3. The authors' use of subjective language, such as "the weight embedding are ingeniously incorporated into node embeddings" and referring to alternative approaches as "clumsy multi-model methods" detracts from the objectivity of the paper. A more impartial tone that directly outlines the methods would enhance the clarity and professionalism of the writing.

[2] Xi Lin, Zhiyuan Yang, and Qingfu Zhang. Pareto set learning for neural multi-objective combinatorial optimization. In International Conference on Learning Representations, 2022a.

A few other suggestions to improve the clarity and ease of reading the paper:

1. It would benefit the reader if it is explained that the solution, represented as a sequence $\pi = \{ \pi_1, \dots, \pi_T \}$, is simply a permutation of the nodes for the Traveling Salesman Problem.
2. $P(\pi | \lambda, s)$ is introduced in Section 3.2, but $s$ is only described in 4.1.
3. “IN” (instance normalization) used in Eq (2) is never explicitly stated anywhere.
4. Discussing the venue of publication in Table 1 is not necessary, in my opinion. Neither is the description of baseline methods as "complex" and the authors' method as "neat". Parameters is misspelled in the column header.

I think it would be beneficial to include more theoretical discussions. For example, the paper mentions that the weighted approach can improve generalization; adding a proof for the generalization bound would make the results more convincing. Additionally, when the number of classes approaches infinity, will this weighting approach converge to the average weight?

1. The authors could a clear definition of what they mean by "neat" in the context of their work, and highlight specific sections where they could elaborate on how their method contrasts with the complexity of existing approaches.
2. In the Methods section, while the structure of the proposed architecture is described, I would like to see a more detailed explanation of why each component is expected to improve performance. Specifically, theoretical justifications for key components such as the addition-based weight embedding and the conditional attention mechanism would be helpful. Some discussion of how the proposed components address specific limitations of previous approaches would also strengthen the work.
3. It would be valuable to see more ablation studies that isolate the contributions of each architectural component. For example, a fair experiment design that solely isolates the effect of the addition-based weight embedding.