Paper: Unsupervised graph neural networks with recurrent features for solving combinatorial optimization problems (ICLR-2024)

Reviewer comment (Novelty):
I'm uncertain about the primary contributions of the paper. Are the key contributions limited to the QRF-GNN architecture and the incorporation of random node features combined with pagerank? Or do they also encompass the formulation of the problem as an unsupervised learning approach?

Author response:
The main contribution of the paper is that the use of recurrent features drastically improves the performance of GNNs in solving Combinatorial Optimization (CO) problems. As a result, GNN obtains highly accurate solutions and can even compete  with the best heuristic algorithms for solving MAX-CUT and graph coloring problems. Please see the general comment where we discuss the novelty and contribution of our work in more detail.

Reviewer comment (Novelty):
If the primary contributions are solely related to the architecture and node features, it's worth noting that while the results appear promising, the method might be viewed as somewhat incremental, especially when compared to existing methods like spectral clustering, which can be considered a basic form of unsupervised learning. In such a case, could you provide insights into any theoretical guarantees, if any exist for your work?

Author response:
We can not agree that contributions might be viewed as somewhat incremental only because they are related to architecture and node features, especially if it contains novelty which allows to significantly outperform existing ones. Generally speaking, the development of a novel unsupervised neural network comes down to the selection of the appropriate architecture, features and loss function. Our work is dedicated to the research in the field of unsupervised graph neural networks, therefore in our experiments we compare QRF-GNN with  1) SOTA unsupervised GNNs, 2) SOTA non-learning based conventional heuristic methods that provide the best solutions on the considered benchmark datasets of the MAX-CUT and graph coloring problems. It would also be interesting for us to compare QRF-GNN with any other unsupervised learning method. However, to the best of our knowledge, there is no a successful application of the spectral clustering method which demonstrates high quality results solving CO problems such as MAX-CUT and coloring.



**In such a case, could you provide insights into any theoretical guarantees, if any exist for your work?**

As well as many other heuristic algorithms for solving hard combinatorial optimization problems, graph neural networks do not provide theoretical guarantees. The performance of the proposed method is evaluated by numerical experiments and comparative analysis with the SOTA methods.

Reviewer comment (Reproducibility):
I have concerns about the reproducibility of this work, as it appears the authors haven't made the code base available. Additionally, they mention that the method lacks stability. How do you determine the best run under these circumstances?

Author response:
In the paper we mention the stochasticity of our method, but we do not mean that it lacks stability. The results can vary with the different seed initializations, and this is the common behavior of any other heuristic algorithm that contains some randomness. This is the case for all the algorithms that appear in our experiments section. Therefore, we follow the conventional experimental design for heuristic algorithms with a stochastic component. It is common to report the best value out of several runs and use it in the comparison with other algorithms [1-3]. In addition, in section B of the appendix, you can see the distribution of results for d-regular graphs and each graph from Gset ("default" refers to QRF-GNN). Speaking about stability, in contrast, we can say that GRF-GNN is more robust and stable than, for example, PI-GNN [1], since it provides high quality results for all problems with the same set of hyperparameters.

Reviewer comment (Reproducibility):
Minor nit - The authors could have elaborated a little on the QUBO formulation and the training details before section 3.

Author response:
Unfortunately, the page limit does not allow for much detail to be included in the main body of the article. We have tried to provide a sufficient number of references to the literature so that the interested reader can gain a good understanding of the issues discussed. We hope that the reader will find sufficient technical details placed in the section with experiments and in the appendix, and we are ready to answer any questions regarding the  reproducibility of the results.

Reviewer comment (Experiments):
Did you'll try to solve QUBO problems other than max cut and coloring?

Author response:
We have tried to solve other problems and found the results promising, but the work is not yet finished and this is the subject of further research.

Reviewer comment (Theory):
The rationale behind certain design choices remains unclear. Questions arise regarding the decision to incorporate the recurrent feature, artificial input features, and parallel graph convolutional layers in lieu of the raw GNN. If the objective is to enhance representation capacity, why not consider a more advanced GNN or GraphFormer design? It is also necessary to establish the relationship between these design choices and their relevance to CO problems.

Author response:
#### •  **W2. The rationale behind certain design choices remains unclear. Questions arise regarding the decision to incorporate the recurrent feature, artificial input features, and parallel graph convolutional layers in lieu of the raw GNN.**

We have tried to answer the question about the importance of recurrence above. Specific features and architecture are probably not optimal and indeed are intended to enhance representation capacity without oversmoothing issues. We tried different combinations and, as can be seen from the ablation study in the appendix, recurrence makes the fundamental contribution, while other modifications are incremental. We also tried different convolutional layers such as GCN and GAT, which showed no additional benefits, so we have reported the results for GraphSAGE.

Reviewer comment (Evaluation):
The evaluation presented in this paper does not convincingly demonstrate QRF-GNN's superiority over heuristic methods. In PI-GNN, Fig. 4 and 5 show its advantages in scalability and computational complexity. Yet these properties are not well verified in this paper.

Author response:
#### •  **W3.The evaluation presented in this paper does not convincingly demonstrate QRF-GNN's superiority over heuristic methods.**

In our experiments we   focused mainly on the comparison between QRF-GNN and existing unsupervised GNNs. As for the heuristics, our intention was to put QRF-GNN in the context of the algorithms that provide the best solutions on the considered benchmark datasets (BLS and TSHEA for Gset, HybridEA for COLOR). In fact, there are dozens of heuristics against which QRF-GNN does demonstrate superiority  (e.g. the heuristics presented in Table 1 in [4] , Table 2 in [5] and Table 1 in [3]). However, we present the comparison with the state-of-the-art methods and show that our algorithm is able to outperform them on some instances. In addition, we also compare our GNN with the strong EO heuristic for Max-Cut, which is able to achieve almost optimal result for d-regular graphs [1], and show that QRF-GNN outperforms it. 
We agree that QRF-GNN does not show the best results among all existing methods on the considered benchmark instances. But we would like to question whether this is a weakness taking into account the ability to solve a wide range of CO problems, the linear scalability and the large gap between the results of best conventional heuristics and previously known unsupervised learning methods. Please see also the general comment explaining the results.

**In PI-GNN, Fig. 4 and 5 show its advantages in scalability and computational complexity. Yet these properties are not well verified in this paper.**

We investigated scalability and computational complexity for large random d-regular graphs with vertex counts ranging from 50 thousand to 1 million and found dependence similar to that in PI-GNN (please see Fig. 2 in the appendix). Unlike in Fig. 4 in [4], we calculated the P-value, since the plot scale for the number of cuts does not allow to see the difference in quality for graphs with different numbers of vertices. Additionally, we raised the question of convergence, which has not been done for PI-GNN. Despite the fact that there were not enough iterations for convergence on large graphs, the P-value was still competitive (please compare Fig. 1 in [1] and Fig. 2b in the appendix) given the near-linear computational complexity. It should also be noted that Figs. 4 and 5 in  [4] as well as Fig. 2 in the appendix refer to the scalability of the algorithms without hyperparameter optimization. Under such conditions PI-GNN performs worse then the simple greedy search, while QRF-GNN provides results close to the global optimal bounds (see Fig.1 in [1] and the answer to Q3).

Reviewer comment (Theory):
Why are recurrent features important for solving CO problems?

Author response:
#### • **W1. The key characteristics of the proposed QRF-GNN, i.e., the unsupervised approach based on QUBO and the recurrent design, have previously been explored in methods like PI-GNN and RUN-CSP [3].    Q1.Why are recurrent features important for solving CO problems?**

The authors of PI-GNN indeed proposed to use QUBO as a loss function, but their results without hyperparameter optimization are worse than solutions of greedy based algorithms with linear time complexity, as shown by in [1][2]. Optimizing the hyperparameters of GNN for each individual example is quite time-consuming and still does not allow the original PI-GNN to achieve results close to QRF-GNN.  

The RUN-CSP algorithm exploits recurrence, but it is different from the one  proposed in our paper. The RUN-CSP uses  the hidden state to update the current state of a node, but the message-passing protocol does not incorporate hidden states of neighbors. It is assumed that the desired information from neighbors of a node was aggregated by the neural network from their hidden states to their current states and thus contributing to the update of the current state of the considered node. As we see from the results in the experiments section, this approach does not yield good quality and the ability of the neural network to aggregate the necessary information is limited.

A key feature of our approach is that we use already predicted nodes classes as hidden recurrent states and then  pass them directly through the message-passing protocol as additional vertex properties. The intuition behind this is as follows. Imagine a Max-Cut task and all vertices at the current step belong to one of the classes. The GNN processes the current vertex and finds that it has the same class as its neighbors. In this case GNN forces a vertex to change its class in order to cut edges with the neighbors and improve the loss value. To summarize, the proposed recurrence is important for solving CO problems, because states of vertices depend on states of nearest neighbors. We will add more details to the article to highlight this point. We have also pointed out this issue in our general comment.

Reviewer comment (Experiments):
Since the methodology mainly focuses on the design of the graph networks, have you tried other advanced GNNs or GraphFormers in hand for better performance of this formulation?

Author response:
#### •  **Q2. Since the methodology mainly focuses on the design of the graph networks, have you tried other advanced GNNs or GraphFormers in hand for better performance of this formulation?**

It is possible that more modern architectures like GraphFormers are able to improve expressive power, but we believe that the recurrent feature should make a significant contribution in this case as well, as it changes the behavior of the iteration process. This could be a subject for future research, but the computational complexity of training must also be taken into account.

Reviewer comment (Experiments):
In Table. 5, why is QRF-GNN more efficient than PI-GNN [1] while the main algorithm pipelines seem similar? It probably comes from the setting of PI-GNN in experiments: "In order to obtain the results of PI-GNN, the authors applied hyperparameter optimization for each graph." However, in my understanding, hyperparameter optimization is not a mandatory request of PI-GNN. It is worth investigating the performance of PI-GNN under the same conditions as QRF-GNN to determine if there is still a speed advantage.

Author response:
#### •  **Q3. In Table. 5, why is QRF-GNN more efficient than PI-GNN [4] while the main algorithm pipelines seem similar? It probably comes from the setting of PI-GNN in experiments: "In order to obtain the results of PI-GNN, the authors applied hyperparameter optimization for each graph." However, in my understanding, hyperparameter optimization is not a mandatory request of PI-GNN. It is worth investigating the performance of PI-GNN under the same conditions as QRF-GNN to determine if there is still a speed advantage.**

Indeed, the difference in runtime arises mainly due to per instance hyperparameters optimization of PI-GNN. It is not mandatory to perform such the optimization. However, the quality of PI-GNN results without optimization may suffer greatly. There are cases where even greedy algorithms perform better [1][2]. We have investigated performance of PI-GNN without per instance optimization also for the coloring problem. We ran the code of PI-GNN presented by the authors on COLOR dataset and citation graphs. We set the hyperparameters in two ways: similar to the  hyperparameters of QRF-GNN (PI-GNN 1) and as an average of reported optimal values obtained during optimization (PI-GNN 2). The averaging was carried out separately for the COLOR dataset and citations graphs since their optimal parameters vary greatly. The results in columns "QRF-GNN" and "PI-GNN" of the following table refer to the Table 3 in the paper. The results in "PI-GNN 1"  and "PI-GNN 2" columns are taken under the same conditions as QRF-GNN (the best value out of 10 runs is presented).

|Graph|QRF-GNN|PI-GNN|PI-GNN 1|PI-GNN 2|
|-----|-------|------|--------|--------|
|anna| 0 | 0 | 1 | 1 |
|jean| 0 | 0 | 0 | 1 |
|myciel5| 0 | 0 | 0 | 0 |
|myciel6| 0 | 0 | 0 | 0 |
|queen5_5| 0 | 0 | 0 | 4 |
|queen6_6| 0 | 0 | 3 | 1 |
|queen7_7| 0 | 0 | 9 | 8 |
|queen8_8| 0 | 1 | 4 | 1 |
|queen9_9| 0 | 1 | 6 | 2 |
|queen8_12| 0 | 0 | 1 | 2 |
|queen11_11| 7 | 17 | 19 | 15 |
|queen13_13| 15 | 26 | 31 | 27 |
|Cora| 0 | 0 | 0 | 90 |
|Citeseer| 0 | 0 | 0 | 50 |
|Pubmed| 0 | 17 | 3 | 1250 |

As can be seen from the table, there is a noticeable drop in overall quality. The algorithm fails to find a solution without violations for graphs, which were successfully colored by using per instance optimization. Therefore we decided to consider the optimization process as part of the PI-GNN algorithm. In order to keep the results of PI-GNN competitive, we place the runtime reported by the authors in [3] in Table 5. The QRF-GNN method does not require optimization of hyperparameters for each instance to obtain accurate results, therefore the table shows its runtime without optimization. Perhaps it is possible to find general hyperparameters that allow PI-GNN to work properly, but the authors of [3] do not provide them and the corresponding runtimes, which forces us to compare with what is given in [3].

Reviewer comment (Novelty):
In Section 5, the statement mentions, "...considering that most of the mentioned methods are specialized for solving specific problems...". In fact, many methods try to propose a general solving framework for solving broad CO problems, such as [2] [3] [4]. Additionally, solving algorithms for NP-complete (NPC) problems inherently possess the potential to solve other CO problems since any NP problem can be reduced to an NPC problem.

Author response:
#### •  **Q4. In Section 5, the statement mentions, "...considering that most of the mentioned methods are specialized for solving specific problems...".**

In the text of our paper, the quote was referring to the heuristics mentioned in the previous sentence: BLS and TSHEA for Max-Cut and HybridEA for graph coloring. Our aim was to highlight that general GNN solver can compete with the best specialized algorithms. 

**In fact, many methods try to propose a general solving framework for solving broad CO problems, such as [1] [2] [3].**
 
The algorithms mentioned by the reviewer are very interesting for study and practical use. However, the methods [2],[3] are supervised, which means that the training set of already labeled (i.e. solved) problems is required. Although the supervised learning framework for solving CO problems is very popular, these methods cannot be considered as autonomous algorithms and the comparison between supervised and unsupervised approaches is methodologically incorrect. The method [1] does not need a training set, and we would be interested in comparing QRF-GNN with this method. However, it is not learning-based algorithm, and in our comparative analysis we mainly focused on unsupervised GNNs. 

**Additionally, solving algorithms for NP-complete (NPC) problems inherently possess the potential to solve other CO problems since any NP problem can be reduced to an NPC problem.**

We doubt the practical application of this statement and have not seen any algorithm based on this in the literature. We would like to ask the reviewer to share examples if they know some. Sometimes it makes sense to formulate one problem in terms of another. For instance, Mixed-Integer Linear Programming  and QUBO formulations have proven its effectiveness in practice. However, this is not directly related to polynomial-time reductions in NP-completeness theory.

Reviewer comment (Novelty):
The paper's novelty appears constrained, utilizing a GNN with QUBO's continuous relaxation as a loss function for combinatorial optimization. Clarification on how this differs from prior work would be beneficial.

Author response:
#### •  **Q1. The paper's novelty appears constrained, utilizing a GNN with QUBO's continuous relaxation as a loss function for combinatorial optimization. Clarification on how this differs from prior work would be beneficial.** 
The main difference from the previous architectures is the use of recurrent node features within the message-passing protocol. We have written a more detailed answer in the general comment, please find additional information there.

#### **Q. Thanks for the response from the authors. I still have questions about the novelty and motivation of the proposed designs. It seems the novelty lies in the use of recurrent node features. However, I didn't see much emphasis on the recurrent node feature in the paper, and the motivation for using recurrent node features is unclear to me.**

Dear reviewer, please allow us to further outline the motivation for using the recurrent feature. The QUBO based loss function is directly determined by the state of vertices relative to the state of their neighbors. For example, in the graph coloring problem, if two adjacent vertices have the same class, i.e. have the same color, then the value of the loss function will proportionally increase by 1. The optimal solution is reached when all neighboring vertices have different colors, in which case the loss function value is zero. The information about what colors are assigned to the neighbors of the current vertex allows the neural network to change the color of this vertex accordingly. To the best of our knowledge, current vertex classes have not previously been used recursively within the message-passing protocol.


If the recurrent feature is removed from the architecture, the quality of the remaining model's results will be comparable to those of simple greedy algorithms. Using only the recurrent feature without one of the parallel layers will already allow to surpass the results of other unsupervised GNNs. After incorporating parallel layers and predefining specific features for nodes (e.g. random vectors, pagerank, and shared vectors), the results on the considered problems can compete with the best specialized heuristics and may even exceed them on certain graphs. All of the these statements are supported by the ablation study in the Appendix.

In Section 3, we describe the mechanism of the recurrent feature as part of QRG-GNN, and in Sections 1,3,5 we try to emphasize its importance, in particular by drawing on numerical experiments and the ablation study.

Reviewer comment (Experiments):
The motivation behind the specific design of the GNN, particularly the use of two different convolutional layers (SAGEConv Pool and SAGEConv mean) in the first layer, is not explained. A rationale for these choices should be provided.

Author response:
#### •  **Q2. The motivation behind the specific design of the GNN, particularly the use of two different convolutional layers (SAGEConv Pool and SAGEConv mean) in the first layer, is not explained. A rationale for these choices should be provided.**
The use of different parallel layers was intended to improve representation capacity while preventing the oversmoothing effect. It should be noted, that the particular design of the architecture plays minor role, since the main gain in quality comes from the recurrence, which is supported by the ablation study in the section B of the appendix.

#### **Also, authors said, "The use of different parallel layers was intended to improve representation capacity while preventing the oversmoothing effect"; "Other modifications of the architecture (i.e. parallel convolutional layers) and different static features were used to increase the expressive power of GNN. " Where do these conclusions come from?**

Different aggregation functions in parallel layers allow to accumulate information of different properties. As the ablation study shows, in some cases it allows to get an advantage over using one layer with one fixed aggregation function, even if we double the number of parameters in this layer. Unfortunately, increasing the depth in graph neural networks is challenging, since it leads to an exponential convergence of vertex features to similar values, and a significant amount of meaningful information is lost. This over-smoothing issue is a known problem in the community [1-3]. 

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[1] T. Konstantin Rusch, Michael M. Bronstein, and Siddhartha Mishra. A survey on over-smoothing in graph neural networks. ArXiv, abs/2303.10993, 2023. 

[2] Oono, K. (2019, May 27). Graph Neural Networks Exponentially Lose Expressive Power for Node Classification. ArXiv.Org. https://arxiv.org/abs/1905.10947

 [3] Chen, D. (2019, September 7), Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View. ArXiv.Org. https://arxiv.org/abs/1909.03211

Reviewer comment (Experiments):
An ablation study is needed to substantiate the proposed GNN's superiority over existing methods. Without this, the advantage of the proposed model remains unclear.

Author response:
#### •  **Q3. An ablation study is needed to substantiate the proposed GNN's superiority over existing methods. Without this, the advantage of the proposed model remains unclear.**
In the appendix of the paper we provide a detailed ablation study of our method, showing the contribution of each part of the algorithm.  To demonstrate the advantage of the proposed GNN over existing methods, numerical experiments are given in section 4 of the main body of the paper. Could you please give us more specific request, what other analyses would be helpful for us to illustrate the advantage of the proposed model?